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Hindawi Publishing CorporationISRNMathematical AnalysisVolume 2013 Article ID 382312 6 pageshttpdxdoiorg1011552013382312
Research ArticleA Subclass of Harmonic Univalent Functions Associated with119902-Analogue of Dziok-Srivastava Operator
Huda Aldweby and Maslina Darus
School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia43600 Bangi Selangor Malaysia
Correspondence should be addressed to Maslina Darus maslinaukmmy
Received 26 June 2013 Accepted 1 August 2013
Academic Editors G Olafsson and D-X Zhou
Copyright copy 2013 H Aldweby and M Darus This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We study a class of complex-valued harmonic univalent functions using a generalized operator involving basic hypergeometricfunction Precisely we give a necessary and sufficient coefficient condition for functions in this class Distortion bounds extremepoints and neighborhood of such functions are also considered
1 Introduction
Let U = 119911 isin C |119911| lt 1 be the open unit disc and let119878119867denote the class of functions which are complex valued
harmonic univalent and sense preserving in U normalizedby 119891(0) = 119891
119911(0) minus 1 = 0 Each 119891 isin 119878
119867can be expressed as
119891 = ℎ + 119892 where ℎ and 119892 are analytic in U We call ℎ theanalytic part and 119892 the coanalytic part of 119891 A necessary andsufficient condition for 119891 to be locally univalent and sensepreserving in U is that |ℎ1015840(119911)| gt |1198921015840(119911)| in U (see [1]) In [2]there is a more comprehensive study on harmonic univalentfunctions Thus for 119891 = ℎ + 119892 isin 119878
119867 we may write
ℎ (119911) = 119911 +
infin
sum
119896=2
119886119896119911
119896 119892 (119911) =
infin
sum
119896=1
119887119896119911
119896(0 le 119887
1lt 1)
(1)
Note that 119878119867reduces to 119878 the class of normalized analytic
univalent functions if the coanalytic part of 119891 = ℎ + 119892 isidentically zero
The study of basic hypergeometric series (also called 119902-hypergeometric series) essentially started in 1748 when Eulerconsidered the infinite product (119902 119902)minus1
infin= prod
infin
119896=0(1 minus 119902
119896+1)
minus1In the literature we were told that the development of thesefunctions was much slower until in 1878 Heine converteda simple observation that lim
119902rarr1[(1 minus 119902
119886)(1 minus 119902)] = 119886
which returns the theory of21206011basic hypergeometric series to
the famous theory of Gaussrsquos21198651hypergeometric series The
importance of basic hypergeometric functions is due to theirapplication in deriving 119902-analogue of well-known functionssuch as 119902-analogues of the exponential gamma and betafunctions In this paper we define a class of starlike har-monic functions using basic hypergeometric functions andinvestigate its properties like coefficient condition distortiontheorem and extreme points
For complex parameters 119886119894 119887119895 119902 (119894 = 1 119903 119895 = 1 119904
119887119895isin C 0 minus1 minus2 |119902| lt 1) we define the basic
hypergeometric function119903Φ119904(1198861 119886
119903 1198871 119887
119904 119902 119911) by
119903Φ119904(1198861 119886119903 1198871 119887
119904 119902 119911)
=
infin
sum
119896=0
(1198861 119902)
119896sdot sdot sdot (119886119903 119902)
119896
(119902 119902)
119896(1198871 119902)
119896sdot sdot sdot (119887119904 119902)
119896
119911
119896
(2)
(119903 = 119904 + 1 119903 119904 isin N0= N cup 0 119911 isin U) where N denote
the set of positive integers and (119886 119902)119896is the 119902-shifted factorial
defined by
(119886 119902)
119896
=
1 119896 = 0
(1 minus 119886) (1 minus 119886119902) (1 minus 119886119902
2) sdot sdot sdot (1 minus 119886119902
119896minus1) 119896 isin N
(3)
2 ISRNMathematical Analysis
We note that
lim119902rarr1
minus
[119903Φ119904(119902
1198861
119902
119886119903
119902
1198871
119902
119887119904
119902 (119902 minus 1)
1+119904minus119903119911)]
=119903119865119904(1198861 119886
119903 1198871 119887
119904 119911)
(4)
where119903119865119904(1198861 119886
119903 1198871 119887
119904 119911) is the well-known general-
ized hypergeometric function By the ratio test one observesthat for |119902| lt 1 and 119903 = 119904 + 1 the series defined in (2)converges absolutely in U so that it represented an analyticfunction in U For more mathematical background of basichypergeometric functions one may refer to [3 4]
The 119902-derivative of a function ℎ(119909) is defined by
119863119902 (ℎ (119909)) =
ℎ (119902119909) minus ℎ (119909)
(119902 minus 1) 119909
119902 = 1 119909 = 0 (5)
For a function ℎ(119911) = 119911119896 we can observe that
119863119902 (ℎ (119911)) = 119863119902
(119911
119896) =
1 minus 119902
119896
1 minus 119902
119911
119896minus1= [119896]119902
119911
119896minus1 (6)
Then lim119902rarr1
119863119902(ℎ(119911)) = lim
119902rarr1[119896]119902119911
119896minus1= 119896119911
119896minus1= ℎ
1015840(119911)
where ℎ1015840(119911) is the ordinary derivative For more properties of119863119902 see [4 5]Corresponding to the function
119903Φ119904(1198861 119886
119903 1198871 119887
119904
119902 119911) consider
119903G119904(1198861 119886
119903 1198871 119887
119904 119902 119911)
= 119911119903Φ119904(1198861 119886
119903 1198871 119887
119904 119902 119911)
= 119911 +
infin
sum
119896=2
(1198861 119902)
119896minus1sdot sdot sdot (119886119903 119902)
119896minus1
(119902 119902)
119896minus1(1198871 119902)
119896minus1sdot sdot sdot (119887119904 119902)
119896minus1
119911
119896
(7)
The authors [6] defined the linear operator 119867119903119904(1198861 119886
119903
1198871 119887
119904 119902)119891 A rarr A by
119867
119903
119904(1198861 119886
119903 1198871 119887
119904 119902) 119891 (119911)
=119903G119904(1198861 119886
119903 1198871 119887
119904 119902 119911) lowast 119891 (119911)
= 119911 +
infin
sum
119896=2
Γ (1198861 119902 119896) 119886
119896119911
119896
(8)
where (lowast) stands for convolution and
Γ (1198861 119902 119896) =
(1198861 119902)
119896minus1sdot sdot sdot (119886119903 119902)
119896minus1
(119902 119902)
119896minus1(1198871 119902)
119896minus1sdot sdot sdot (119887119904 119902)
119896minus1
(9)
To make the notation simple we write
119867
119903
119904[1198861 119902] 119891 (119911) = 119867
119903
119904(1198861 119886
119903 1198871 119887
119904 119902) 119891 (119911) (10)
We define the operator (8) of harmonic function 119891 = ℎ + 119892given by (1) as
119867
119903
119904[1198861 119902] 119891 (119911) = 119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911) (11)
Definition 1 For 0 le 120575 lt 1 let 119878lowast119867(1198861 120575 119902) denote the
subfamily of starlike harmonic functions 119891 isin 119878lowast119867of the form
(1) such that
120597
120597120579
(arg119867119903119904[1198861 119902] 119891) ge 120575 |119911| = 119903 lt 1 (12)
Following [7] a function 119891 is said to be in the class119881119867(1198861 120575 119902) = 119878
lowast
119867(1198861 120575 119902) cap 119881
119867if 119891 of the form (1) satisfies
the condition that
arg (119886119896) = 120579119896
arg (119887119896) = 120599119896 (119896 ge 119899 + 1 119899 isin N) (13)
and if there exists a real number 120588 such that
120579119896+ (119896 minus 1) 120601 equiv 120587 (mod 2120587) 120599
119896+ (119896 minus 1) 120601 equiv 0
(119896 ge 119899 + 1 119899 isin N) (14)
By specializing the parameters of 119867119903119904[1198861 119902]119891 we obtain
different classes of starlike harmonic functions for example
(i) for 119903 = 119904 + 1 1198862= 1198871 119886
119903= 119887119904 119878lowast119867(119902 119902 120575) =
119878119867(120575) [8] is the class of sense-preserving harmonicunivalent functions 119891 which are starlike of order 120575 inU that is 120597120597120579(arg119891(119903119890i120579)) ge 120575
(ii) for 119903 = 119904 + 1 1198862= 1198871 119886
119903= 119887119904 and 119886
1=
119902
119899+1 119902 rarr 1 119878lowast
119867(119902
119899+1 119902 120575) = 119877
119867(119899 120572) [9] is the
class of starlike harmonic univalent functions with(120597120597120579)(arg119863119899119891(119911)) ge 120575 where 119863 is the Ruscheweyhderivative (see [10])
(iii) for 119894 = 1 119903 119895 = 1 119904 119903 = 119904 + 1 119886119894= 119902
120572119894
and 119887119895= 119902
120573119895 119902 rarr 1 119878lowast
119867(1198861 119902 120575) = 119878
lowast
119867(1205721 120575) [11]
is the class of starlike harmonic univalent functionswith (120597120597120579)(arg119867119903
119904[1205721]119891) ge 120575 where 119867119903
119904[1205721] is the
Dziok-Srivastava operator (see [12])
2 Main Results
In our first theorem we introduce a sufficient coefficientbound for harmonic functions in 119878lowast
119867(1198861 120575 119902)
Theorem 2 Let 119891 = ℎ + 119892 be given by (1) If
infin
sum
119896=2
(
[119896]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[119896]119902+ 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896)
le 1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
(15)
where 1198861= 1 0 le 120575 lt 1 and Γ(119886
1 119902 119896) is given by (9) then
119891 isin 119878
lowast
119867(1198861 120575 119902)
ISRNMathematical Analysis 3
Proof To prove that 119891 isin 119878lowast119867(1198861 120575 119902) we only need to show
that if (15) holds then the required condition (12) is satisfiedFor (12) we can write
120597
120597120579
(arg119867119903119904[1198861 119902] 119891 (119911))
= R
119911119863119902(119867
119903
119904[1198861 119902] ℎ (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus
119911119863119902(119867
119903
119904[1198861 119902] 119892 (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
= R119860 (119911)
119861 (119911)
(16)
Using the fact that R(119908) ge 120575 if and only if |1 minus 120575 + 119908| ge|1 + 120575 minus 119908| it suffices to show that
|119860 (119911) + (1 minus 120575) 119861 (119911)| minus |119860 (119911) minus (1 + 120575) 119861 (119911)| ge 0 (17)
Substituting for 119860(119911) and 119861(119911) in (15) yields
|119860 (119911) + (1 minus 120575) 119861 (119911)| minus |119860 (119911) minus (1 + 120575) 119861 (119911)|
ge (2 minus 120575) |119911| minus
infin
sum
119896=2
([119896]119902+ 1 minus 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
|119911|
119896
minus
infin
sum
119896=1
([119896]119902minus 1 + 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
|119911|
119896
minus 120575 |119911| minus
infin
sum
119896=2
([119896]119902minus 1 minus 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
|119911|
119896
minus
infin
sum
119896=1
([119896]119902+ 1 + 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
|119911|
119896
ge 2 (1 minus 120575) |119911| 1 minus
infin
sum
119896=2
[119896]119902minus 120575
1 minus 120575
Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
minus
infin
sum
119896=1
[119896]119902+ 120575
1 minus 120575
Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
= 2 (1 minus 120575) |119911| 1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
minus [
infin
sum
119896=2
(
[119896]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[119896]119902+ 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
)]
times Γ (1198861 119902 119896)
(18)
The last expression is nonnegative by (15) and so 119891 isin
119878
lowast
119867(1198861 120575 119902)
Nowwe obtain the necessary and sufficient conditions for119891 = ℎ + g given by (14)
Theorem 3 Let 119891 = ℎ + 119892 be given by (11) Then 119891 isin
119881
119867(1198861 120575 119902) if and only if
infin
sum
119896=2
(
[119896]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[119896]119902+ 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896)
le 1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
(19)
where 1198861= 1 0 le 120575 lt 1 and Γ(119886
1 119902 119896) is given by (9)
Proof Since 119881
119867(1198861 120575 119902) sub 119878
lowast
119867(1198861 120575 119902) we only to
prove the only if part of the theorem So that for func-tions 119891 isin 119881
119867(1198861 120575 119902) we notice that the condition
(120597120597120579)(arg119867119903119904[1198861 119902]119891(119911)) ge 120575 is equivalent to
120597
120597120579
(arg119867119903119904[1198861 119902] 119891 (119911)) minus 120575
= R119911119863119902(119867
119903
119904[1198861 119902] ℎ (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus
119911119863119902(119867
119903
119904[1198861 119902] 119892 (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus120575 ge 0
(20)
That is
R[
[
(1 minus 120575) 119911 + sum
infin
119896=2([119896]119902
minus 120575) Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
119911
119896minus sum
infin
119896=1([119896]119902
+ 120575) Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
119911
119896
119911 + sum
infin
119896=2Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
119911
119896+ sum
infin
119896=1Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
119911
119896
]
]
ge 0 (21)
4 ISRNMathematical Analysis
The previous condition must hold for all values of 119911 in UUpon choosing 120601 according to (14) we must have
(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
+ sum
infin
119896=2(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
minus
sum
infin
119896=2(([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
+ sum
infin
119896=2(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
ge 0
(22)
If condition (19) does not hold then the numerator in (22)is negative for 119903 sufficiently close to 1 Hence there exist1199110= 1199030in (01) for which the quotient of (22) is negativeThis
contradicts the fact that 119891 isin 119881119867(1198861 120575 119902) and this completes
the proof
The following theorem gives the distortion bounds forfunctions in119881
119867(1198861 120575 119902)which yield a covering result for this
class
Theorem 4 If 119891 isin 119881119867(1198861 120575 119902) then
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
+
1
Γ (1198861 119902 2)
(
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
) 119903
2
|119911| = 119903 lt 1
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
ge (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
minus
1
Γ (1198861 119902 2)
(
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
) 119903
2
|119911| = 119903 lt 1
(23)
where
Γ (1198861 119902 2) =
(1 minus 1198861) sdot sdot sdot (1 minus 119886
119903)
(1 minus 119902) (1 minus 1198871) sdot sdot sdot (1 minus 119887
119904)
[2]119902= (119902 + 1)
(24)
Proof Wewill only prove the right hand inequalityThe prooffor the left hand inequality is similar
let 119891 isin 119881119867(1198861 120575 119902) Taking the absolute value of 119891 we
obtain
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
infin
sum
119896=2
(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) 119903
119896
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
infin
sum
119896=2
(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) 119903
2
(25)
That is
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1 minus 120575
Γ (1198861 119902 2) ([2]119902
minus 120575)
times
infin
sum
119896=2
(
[2]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[2]119902minus 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
)
times Γ (1198861 119902 2) 119903
2
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1 minus 120575
Γ (1198861 119902 2) ((119902 + 1) minus 120575)
times [1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
] 119903
2
= (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1
Γ (1198861 119902 2)
times [
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
2
(26)
Corollary 5 Let 119891 be of the form (1) so that 119891 isin 119881119867(1198861 120575 119902)
Then
119908 |119908| lt
2Γ (1198861 119902 2) minus 1 minus (Γ (119886
1 119902 2) minus 1) 120575
((119902 + 1) minus 120575) Γ (1198861 119902 2)
minus
2Γ (1198861 119902 2) minus 1 minus (Γ (119886
1 119902 2) minus 1) 120575
((119902 + 1) + 120575) Γ (1198861 119902 2)
1003816100381610038161003816
1198871
1003816100381610038161003816
sub 119891 (U)
(27)
Next one determines the extreme points of closed convex hullof 119881119867(1198861 120575 119902) denoted by clco119881
119867(1198861 120575 119902)
Theorem 6 Set
120582119896=
1 minus 120575
([119896]119902minus 120575) Γ (119886
1 119902 119896)
120583119896=
1 minus 120575
([119896]119902+ 120575) Γ (119886
1 119902 119896)
(28)
For 1198871fixed the extreme points for clco119881
119867(1198861 120575 119902) are
119911 + 120582119896119909119911
119896+ 1198871119911 cup 119911 + 119887
1119911 + 120583119896119909119911
119896 (29)
where 119896 ge 2 and |119909| = 1 minus |1198871|
Proof Any function 119891 isin clco119881119867(1198861 120575 119902) may be expressed
as
119891 (119911) = 119911 +
infin
sum
119896=2
1003816100381610038161003816
119886119896
1003816100381610038161003816
119890
119894120572119896
119911
119896+ 1198871119911 +
infin
sum
119896=2
1003816100381610038161003816
119887119896
1003816100381610038161003816
119890
119894120573119896
119911
119896 (30)
ISRNMathematical Analysis 5
where the coefficients satisfy the inequality (15) Set ℎ1(119911) =
119911 1198921(119911) = 119887
1119911 ℎ119896(119911) = 119911 + 120582
119896119890
119894120572119896
119911
119896 and 119892119896(119911) = 119887
1(119911) +
119890
119894120573119896
119911
119896 for 119896 = 2 3 Writing 119883119896= |119886119896|120582119896 119884119896= |119887119896|120583119896
119896 = 2 3 and1198831= 1minussum
infin
119896=2119883119896 1198841= 1minussum
infin
119896=2119884119896 we have
119891 (119911) =
infin
sum
119896=1
(119883119896ℎ119896 (119911) + 119884119896
119892119896 (119911)) (31)
In particular set
1198911 (119911) = 119911 + 1198871
119911 119891119896 (119911) = 119911 + 120582119896
119909119911
119896+ 1198871119911 + 120583119896119910119911
119896
(119896 ge 2 |119909| +
1003816100381610038161003816
119910
1003816100381610038161003816
= 1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(32)
Therefore the extreme points of clco119881119867(1198861 120575 119902) are
contained in 119891119896(119911) To see that 119891
1is not an extreme point
note that1198911may be written as a convex linear combination of
functions in clco119881119867(1198861 120575 119902) as follows
1198911 (119911) =
1
2
1198911 (119911) + 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
+
1
2
1198911 (119911) minus 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
(33)
If both |119909| = 0 and |119910| = 0 we will show that it can also beexpressed as a convex linear combination of functions inclco119881
119867(1198861 120575 119902) Without loss of generality assume that |119909| ge
|119910| Choose 120598 gt 0 small enough so that 120598 lt |119909||119910| Set119860 = 1 + 120598 and 119861 = 1 minus |120598119909119910| We then see that both
1199051 (119911) = 119911 + 120582119896
119860119909119911
119896+ 1198871119911 + 120583119896119910119861119911
119896
1199052 (119911) = 119911 + 120582119896 (
2 minus 119860) 119909119911
119896+ 1198871119911 + 120583119896119910 (2 minus 119861) 119911
119896
(34)
are in clco119881119867(1198861 120575 119902) and note that
119891119899 (119911) =
1
2
1199051 (119911) + 1199052 (
119911) (35)
The extremal coefficient bound shows that the functions ofthe form (29) are extreme points for clco119881
119867(1198861 120575 119902) and so
the proof is complete
Following the earlier works in [2 13] we refer to the 120574-neighborhood of the functions 119891 defined by (1) to be the setof functions 119865 for which
119873120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
119896 (
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
) +
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(36)
We define the 119902-120574-neighborhood of a function 119891 isin 119878119867as
follows
119873
119902
120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
[119896]119902(
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
)
+
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(37)
In our case let us define the generalized 119902-120574-neighborhood of 119891 to be the set
119873
119902
120574(119891) = 119865
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
]
+ (1 + 120575)
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le (1 minus 120575) 120574
(38)
Theorem 7 Let 119891 be given by (1) If 119891 satisfies the conditions
infin
sum
119896=2
[119896]119902([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
Γ (1198861 119902 119896)
+
infin
sum
1=2
[119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
Γ (1198861 119902 119896) le 1 minus 120575
0 le 120575 lt 1
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(39)
then119873119902120574(119891) sub 119878
lowast
119867(1198861 120575 119902)
Proof Let 119891 satisfy (39) and let
119865 (119911) = 119911 + 1198611119911 +
infin
sum
119896=2
(119860119896119911
119896+ 119861119896119911
119896) (40)
belong to119873119902120574(119891) We have
(1 + 120575)
1003816100381610038161003816
1198611
1003816100381610038161003816
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896
1003816100381610038161003816
]
le (1 + 120575)
1003816100381610038161003816
1198611minus 1198871
1003816100381610038161003816
+ (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
6 ISRNMathematical Analysis
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896minus 119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896minus 119887119896
1003816100381610038161003816
]
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
[2]119902minus 120575
infin
sum
119896=2
Γ (1198861 119902 119896) [[119896]119902
([119896]119902minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ [119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
(119902 + 1) minus 120575
times [(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
] le 1 minus 120575
(41)
Hence
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
119865 isin 119878
lowast
119867(1198861 120575 119902)
(42)
Acknowledgment
The work presented here was partially supported by LRGSTD2011UKMICT0302 and UKM-DLP-2011-050
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A vol 9 pp 3ndash251984
[2] Y Avcı and E Złotkiewicz ldquoOn harmonic univalent mappingsrdquoAnnales Universitatis Mariae Curie-Skłodowska A vol 44 pp1ndash7 1990
[3] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[4] H Exton q-hypergeometric functions and applications EllisHorwood Series Mathematics and its Applications Ellis Hor-wood Chichester UK 1983
[5] H A Ghany ldquo119902-derivative of basic hypergeometric series withrespect to parametersrdquo International Journal of MathematicalAnalysis vol 3 no 33-36 pp 1617ndash1632 2009
[6] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquoMatematicki Vesnik vol 65no 4 pp 454ndash465 2013
[7] J M Jahangiri and H Silverman ldquoHarmonic univalent func-tions with varying argumentsrdquo International Journal of AppliedMathematics vol 8 no 3 pp 267ndash275 2002
[8] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[9] G Murugusundaramoorthy ldquoA class of Ruscheweyh-type har-monic univalent functions with varying argumentsrdquo SouthwestJournal of Pure and AppliedMathematics no 2 pp 90ndash95 2003
[10] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[11] H A Al-Kharsani and R A Al-Khal ldquoUnivalent harmonicfunctionsrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 8 no 2 article 59 p 8 2007
[12] J Dziok and H M Srivastava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
[13] S Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society vol 81 no 4 pp521ndash527 1981
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 ISRNMathematical Analysis
We note that
lim119902rarr1
minus
[119903Φ119904(119902
1198861
119902
119886119903
119902
1198871
119902
119887119904
119902 (119902 minus 1)
1+119904minus119903119911)]
=119903119865119904(1198861 119886
119903 1198871 119887
119904 119911)
(4)
where119903119865119904(1198861 119886
119903 1198871 119887
119904 119911) is the well-known general-
ized hypergeometric function By the ratio test one observesthat for |119902| lt 1 and 119903 = 119904 + 1 the series defined in (2)converges absolutely in U so that it represented an analyticfunction in U For more mathematical background of basichypergeometric functions one may refer to [3 4]
The 119902-derivative of a function ℎ(119909) is defined by
119863119902 (ℎ (119909)) =
ℎ (119902119909) minus ℎ (119909)
(119902 minus 1) 119909
119902 = 1 119909 = 0 (5)
For a function ℎ(119911) = 119911119896 we can observe that
119863119902 (ℎ (119911)) = 119863119902
(119911
119896) =
1 minus 119902
119896
1 minus 119902
119911
119896minus1= [119896]119902
119911
119896minus1 (6)
Then lim119902rarr1
119863119902(ℎ(119911)) = lim
119902rarr1[119896]119902119911
119896minus1= 119896119911
119896minus1= ℎ
1015840(119911)
where ℎ1015840(119911) is the ordinary derivative For more properties of119863119902 see [4 5]Corresponding to the function
119903Φ119904(1198861 119886
119903 1198871 119887
119904
119902 119911) consider
119903G119904(1198861 119886
119903 1198871 119887
119904 119902 119911)
= 119911119903Φ119904(1198861 119886
119903 1198871 119887
119904 119902 119911)
= 119911 +
infin
sum
119896=2
(1198861 119902)
119896minus1sdot sdot sdot (119886119903 119902)
119896minus1
(119902 119902)
119896minus1(1198871 119902)
119896minus1sdot sdot sdot (119887119904 119902)
119896minus1
119911
119896
(7)
The authors [6] defined the linear operator 119867119903119904(1198861 119886
119903
1198871 119887
119904 119902)119891 A rarr A by
119867
119903
119904(1198861 119886
119903 1198871 119887
119904 119902) 119891 (119911)
=119903G119904(1198861 119886
119903 1198871 119887
119904 119902 119911) lowast 119891 (119911)
= 119911 +
infin
sum
119896=2
Γ (1198861 119902 119896) 119886
119896119911
119896
(8)
where (lowast) stands for convolution and
Γ (1198861 119902 119896) =
(1198861 119902)
119896minus1sdot sdot sdot (119886119903 119902)
119896minus1
(119902 119902)
119896minus1(1198871 119902)
119896minus1sdot sdot sdot (119887119904 119902)
119896minus1
(9)
To make the notation simple we write
119867
119903
119904[1198861 119902] 119891 (119911) = 119867
119903
119904(1198861 119886
119903 1198871 119887
119904 119902) 119891 (119911) (10)
We define the operator (8) of harmonic function 119891 = ℎ + 119892given by (1) as
119867
119903
119904[1198861 119902] 119891 (119911) = 119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911) (11)
Definition 1 For 0 le 120575 lt 1 let 119878lowast119867(1198861 120575 119902) denote the
subfamily of starlike harmonic functions 119891 isin 119878lowast119867of the form
(1) such that
120597
120597120579
(arg119867119903119904[1198861 119902] 119891) ge 120575 |119911| = 119903 lt 1 (12)
Following [7] a function 119891 is said to be in the class119881119867(1198861 120575 119902) = 119878
lowast
119867(1198861 120575 119902) cap 119881
119867if 119891 of the form (1) satisfies
the condition that
arg (119886119896) = 120579119896
arg (119887119896) = 120599119896 (119896 ge 119899 + 1 119899 isin N) (13)
and if there exists a real number 120588 such that
120579119896+ (119896 minus 1) 120601 equiv 120587 (mod 2120587) 120599
119896+ (119896 minus 1) 120601 equiv 0
(119896 ge 119899 + 1 119899 isin N) (14)
By specializing the parameters of 119867119903119904[1198861 119902]119891 we obtain
different classes of starlike harmonic functions for example
(i) for 119903 = 119904 + 1 1198862= 1198871 119886
119903= 119887119904 119878lowast119867(119902 119902 120575) =
119878119867(120575) [8] is the class of sense-preserving harmonicunivalent functions 119891 which are starlike of order 120575 inU that is 120597120597120579(arg119891(119903119890i120579)) ge 120575
(ii) for 119903 = 119904 + 1 1198862= 1198871 119886
119903= 119887119904 and 119886
1=
119902
119899+1 119902 rarr 1 119878lowast
119867(119902
119899+1 119902 120575) = 119877
119867(119899 120572) [9] is the
class of starlike harmonic univalent functions with(120597120597120579)(arg119863119899119891(119911)) ge 120575 where 119863 is the Ruscheweyhderivative (see [10])
(iii) for 119894 = 1 119903 119895 = 1 119904 119903 = 119904 + 1 119886119894= 119902
120572119894
and 119887119895= 119902
120573119895 119902 rarr 1 119878lowast
119867(1198861 119902 120575) = 119878
lowast
119867(1205721 120575) [11]
is the class of starlike harmonic univalent functionswith (120597120597120579)(arg119867119903
119904[1205721]119891) ge 120575 where 119867119903
119904[1205721] is the
Dziok-Srivastava operator (see [12])
2 Main Results
In our first theorem we introduce a sufficient coefficientbound for harmonic functions in 119878lowast
119867(1198861 120575 119902)
Theorem 2 Let 119891 = ℎ + 119892 be given by (1) If
infin
sum
119896=2
(
[119896]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[119896]119902+ 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896)
le 1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
(15)
where 1198861= 1 0 le 120575 lt 1 and Γ(119886
1 119902 119896) is given by (9) then
119891 isin 119878
lowast
119867(1198861 120575 119902)
ISRNMathematical Analysis 3
Proof To prove that 119891 isin 119878lowast119867(1198861 120575 119902) we only need to show
that if (15) holds then the required condition (12) is satisfiedFor (12) we can write
120597
120597120579
(arg119867119903119904[1198861 119902] 119891 (119911))
= R
119911119863119902(119867
119903
119904[1198861 119902] ℎ (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus
119911119863119902(119867
119903
119904[1198861 119902] 119892 (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
= R119860 (119911)
119861 (119911)
(16)
Using the fact that R(119908) ge 120575 if and only if |1 minus 120575 + 119908| ge|1 + 120575 minus 119908| it suffices to show that
|119860 (119911) + (1 minus 120575) 119861 (119911)| minus |119860 (119911) minus (1 + 120575) 119861 (119911)| ge 0 (17)
Substituting for 119860(119911) and 119861(119911) in (15) yields
|119860 (119911) + (1 minus 120575) 119861 (119911)| minus |119860 (119911) minus (1 + 120575) 119861 (119911)|
ge (2 minus 120575) |119911| minus
infin
sum
119896=2
([119896]119902+ 1 minus 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
|119911|
119896
minus
infin
sum
119896=1
([119896]119902minus 1 + 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
|119911|
119896
minus 120575 |119911| minus
infin
sum
119896=2
([119896]119902minus 1 minus 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
|119911|
119896
minus
infin
sum
119896=1
([119896]119902+ 1 + 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
|119911|
119896
ge 2 (1 minus 120575) |119911| 1 minus
infin
sum
119896=2
[119896]119902minus 120575
1 minus 120575
Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
minus
infin
sum
119896=1
[119896]119902+ 120575
1 minus 120575
Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
= 2 (1 minus 120575) |119911| 1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
minus [
infin
sum
119896=2
(
[119896]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[119896]119902+ 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
)]
times Γ (1198861 119902 119896)
(18)
The last expression is nonnegative by (15) and so 119891 isin
119878
lowast
119867(1198861 120575 119902)
Nowwe obtain the necessary and sufficient conditions for119891 = ℎ + g given by (14)
Theorem 3 Let 119891 = ℎ + 119892 be given by (11) Then 119891 isin
119881
119867(1198861 120575 119902) if and only if
infin
sum
119896=2
(
[119896]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[119896]119902+ 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896)
le 1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
(19)
where 1198861= 1 0 le 120575 lt 1 and Γ(119886
1 119902 119896) is given by (9)
Proof Since 119881
119867(1198861 120575 119902) sub 119878
lowast
119867(1198861 120575 119902) we only to
prove the only if part of the theorem So that for func-tions 119891 isin 119881
119867(1198861 120575 119902) we notice that the condition
(120597120597120579)(arg119867119903119904[1198861 119902]119891(119911)) ge 120575 is equivalent to
120597
120597120579
(arg119867119903119904[1198861 119902] 119891 (119911)) minus 120575
= R119911119863119902(119867
119903
119904[1198861 119902] ℎ (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus
119911119863119902(119867
119903
119904[1198861 119902] 119892 (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus120575 ge 0
(20)
That is
R[
[
(1 minus 120575) 119911 + sum
infin
119896=2([119896]119902
minus 120575) Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
119911
119896minus sum
infin
119896=1([119896]119902
+ 120575) Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
119911
119896
119911 + sum
infin
119896=2Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
119911
119896+ sum
infin
119896=1Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
119911
119896
]
]
ge 0 (21)
4 ISRNMathematical Analysis
The previous condition must hold for all values of 119911 in UUpon choosing 120601 according to (14) we must have
(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
+ sum
infin
119896=2(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
minus
sum
infin
119896=2(([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
+ sum
infin
119896=2(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
ge 0
(22)
If condition (19) does not hold then the numerator in (22)is negative for 119903 sufficiently close to 1 Hence there exist1199110= 1199030in (01) for which the quotient of (22) is negativeThis
contradicts the fact that 119891 isin 119881119867(1198861 120575 119902) and this completes
the proof
The following theorem gives the distortion bounds forfunctions in119881
119867(1198861 120575 119902)which yield a covering result for this
class
Theorem 4 If 119891 isin 119881119867(1198861 120575 119902) then
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
+
1
Γ (1198861 119902 2)
(
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
) 119903
2
|119911| = 119903 lt 1
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
ge (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
minus
1
Γ (1198861 119902 2)
(
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
) 119903
2
|119911| = 119903 lt 1
(23)
where
Γ (1198861 119902 2) =
(1 minus 1198861) sdot sdot sdot (1 minus 119886
119903)
(1 minus 119902) (1 minus 1198871) sdot sdot sdot (1 minus 119887
119904)
[2]119902= (119902 + 1)
(24)
Proof Wewill only prove the right hand inequalityThe prooffor the left hand inequality is similar
let 119891 isin 119881119867(1198861 120575 119902) Taking the absolute value of 119891 we
obtain
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
infin
sum
119896=2
(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) 119903
119896
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
infin
sum
119896=2
(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) 119903
2
(25)
That is
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1 minus 120575
Γ (1198861 119902 2) ([2]119902
minus 120575)
times
infin
sum
119896=2
(
[2]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[2]119902minus 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
)
times Γ (1198861 119902 2) 119903
2
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1 minus 120575
Γ (1198861 119902 2) ((119902 + 1) minus 120575)
times [1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
] 119903
2
= (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1
Γ (1198861 119902 2)
times [
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
2
(26)
Corollary 5 Let 119891 be of the form (1) so that 119891 isin 119881119867(1198861 120575 119902)
Then
119908 |119908| lt
2Γ (1198861 119902 2) minus 1 minus (Γ (119886
1 119902 2) minus 1) 120575
((119902 + 1) minus 120575) Γ (1198861 119902 2)
minus
2Γ (1198861 119902 2) minus 1 minus (Γ (119886
1 119902 2) minus 1) 120575
((119902 + 1) + 120575) Γ (1198861 119902 2)
1003816100381610038161003816
1198871
1003816100381610038161003816
sub 119891 (U)
(27)
Next one determines the extreme points of closed convex hullof 119881119867(1198861 120575 119902) denoted by clco119881
119867(1198861 120575 119902)
Theorem 6 Set
120582119896=
1 minus 120575
([119896]119902minus 120575) Γ (119886
1 119902 119896)
120583119896=
1 minus 120575
([119896]119902+ 120575) Γ (119886
1 119902 119896)
(28)
For 1198871fixed the extreme points for clco119881
119867(1198861 120575 119902) are
119911 + 120582119896119909119911
119896+ 1198871119911 cup 119911 + 119887
1119911 + 120583119896119909119911
119896 (29)
where 119896 ge 2 and |119909| = 1 minus |1198871|
Proof Any function 119891 isin clco119881119867(1198861 120575 119902) may be expressed
as
119891 (119911) = 119911 +
infin
sum
119896=2
1003816100381610038161003816
119886119896
1003816100381610038161003816
119890
119894120572119896
119911
119896+ 1198871119911 +
infin
sum
119896=2
1003816100381610038161003816
119887119896
1003816100381610038161003816
119890
119894120573119896
119911
119896 (30)
ISRNMathematical Analysis 5
where the coefficients satisfy the inequality (15) Set ℎ1(119911) =
119911 1198921(119911) = 119887
1119911 ℎ119896(119911) = 119911 + 120582
119896119890
119894120572119896
119911
119896 and 119892119896(119911) = 119887
1(119911) +
119890
119894120573119896
119911
119896 for 119896 = 2 3 Writing 119883119896= |119886119896|120582119896 119884119896= |119887119896|120583119896
119896 = 2 3 and1198831= 1minussum
infin
119896=2119883119896 1198841= 1minussum
infin
119896=2119884119896 we have
119891 (119911) =
infin
sum
119896=1
(119883119896ℎ119896 (119911) + 119884119896
119892119896 (119911)) (31)
In particular set
1198911 (119911) = 119911 + 1198871
119911 119891119896 (119911) = 119911 + 120582119896
119909119911
119896+ 1198871119911 + 120583119896119910119911
119896
(119896 ge 2 |119909| +
1003816100381610038161003816
119910
1003816100381610038161003816
= 1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(32)
Therefore the extreme points of clco119881119867(1198861 120575 119902) are
contained in 119891119896(119911) To see that 119891
1is not an extreme point
note that1198911may be written as a convex linear combination of
functions in clco119881119867(1198861 120575 119902) as follows
1198911 (119911) =
1
2
1198911 (119911) + 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
+
1
2
1198911 (119911) minus 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
(33)
If both |119909| = 0 and |119910| = 0 we will show that it can also beexpressed as a convex linear combination of functions inclco119881
119867(1198861 120575 119902) Without loss of generality assume that |119909| ge
|119910| Choose 120598 gt 0 small enough so that 120598 lt |119909||119910| Set119860 = 1 + 120598 and 119861 = 1 minus |120598119909119910| We then see that both
1199051 (119911) = 119911 + 120582119896
119860119909119911
119896+ 1198871119911 + 120583119896119910119861119911
119896
1199052 (119911) = 119911 + 120582119896 (
2 minus 119860) 119909119911
119896+ 1198871119911 + 120583119896119910 (2 minus 119861) 119911
119896
(34)
are in clco119881119867(1198861 120575 119902) and note that
119891119899 (119911) =
1
2
1199051 (119911) + 1199052 (
119911) (35)
The extremal coefficient bound shows that the functions ofthe form (29) are extreme points for clco119881
119867(1198861 120575 119902) and so
the proof is complete
Following the earlier works in [2 13] we refer to the 120574-neighborhood of the functions 119891 defined by (1) to be the setof functions 119865 for which
119873120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
119896 (
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
) +
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(36)
We define the 119902-120574-neighborhood of a function 119891 isin 119878119867as
follows
119873
119902
120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
[119896]119902(
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
)
+
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(37)
In our case let us define the generalized 119902-120574-neighborhood of 119891 to be the set
119873
119902
120574(119891) = 119865
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
]
+ (1 + 120575)
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le (1 minus 120575) 120574
(38)
Theorem 7 Let 119891 be given by (1) If 119891 satisfies the conditions
infin
sum
119896=2
[119896]119902([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
Γ (1198861 119902 119896)
+
infin
sum
1=2
[119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
Γ (1198861 119902 119896) le 1 minus 120575
0 le 120575 lt 1
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(39)
then119873119902120574(119891) sub 119878
lowast
119867(1198861 120575 119902)
Proof Let 119891 satisfy (39) and let
119865 (119911) = 119911 + 1198611119911 +
infin
sum
119896=2
(119860119896119911
119896+ 119861119896119911
119896) (40)
belong to119873119902120574(119891) We have
(1 + 120575)
1003816100381610038161003816
1198611
1003816100381610038161003816
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896
1003816100381610038161003816
]
le (1 + 120575)
1003816100381610038161003816
1198611minus 1198871
1003816100381610038161003816
+ (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
6 ISRNMathematical Analysis
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896minus 119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896minus 119887119896
1003816100381610038161003816
]
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
[2]119902minus 120575
infin
sum
119896=2
Γ (1198861 119902 119896) [[119896]119902
([119896]119902minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ [119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
(119902 + 1) minus 120575
times [(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
] le 1 minus 120575
(41)
Hence
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
119865 isin 119878
lowast
119867(1198861 120575 119902)
(42)
Acknowledgment
The work presented here was partially supported by LRGSTD2011UKMICT0302 and UKM-DLP-2011-050
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A vol 9 pp 3ndash251984
[2] Y Avcı and E Złotkiewicz ldquoOn harmonic univalent mappingsrdquoAnnales Universitatis Mariae Curie-Skłodowska A vol 44 pp1ndash7 1990
[3] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[4] H Exton q-hypergeometric functions and applications EllisHorwood Series Mathematics and its Applications Ellis Hor-wood Chichester UK 1983
[5] H A Ghany ldquo119902-derivative of basic hypergeometric series withrespect to parametersrdquo International Journal of MathematicalAnalysis vol 3 no 33-36 pp 1617ndash1632 2009
[6] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquoMatematicki Vesnik vol 65no 4 pp 454ndash465 2013
[7] J M Jahangiri and H Silverman ldquoHarmonic univalent func-tions with varying argumentsrdquo International Journal of AppliedMathematics vol 8 no 3 pp 267ndash275 2002
[8] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[9] G Murugusundaramoorthy ldquoA class of Ruscheweyh-type har-monic univalent functions with varying argumentsrdquo SouthwestJournal of Pure and AppliedMathematics no 2 pp 90ndash95 2003
[10] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[11] H A Al-Kharsani and R A Al-Khal ldquoUnivalent harmonicfunctionsrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 8 no 2 article 59 p 8 2007
[12] J Dziok and H M Srivastava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
[13] S Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society vol 81 no 4 pp521ndash527 1981
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRNMathematical Analysis 3
Proof To prove that 119891 isin 119878lowast119867(1198861 120575 119902) we only need to show
that if (15) holds then the required condition (12) is satisfiedFor (12) we can write
120597
120597120579
(arg119867119903119904[1198861 119902] 119891 (119911))
= R
119911119863119902(119867
119903
119904[1198861 119902] ℎ (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus
119911119863119902(119867
119903
119904[1198861 119902] 119892 (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
= R119860 (119911)
119861 (119911)
(16)
Using the fact that R(119908) ge 120575 if and only if |1 minus 120575 + 119908| ge|1 + 120575 minus 119908| it suffices to show that
|119860 (119911) + (1 minus 120575) 119861 (119911)| minus |119860 (119911) minus (1 + 120575) 119861 (119911)| ge 0 (17)
Substituting for 119860(119911) and 119861(119911) in (15) yields
|119860 (119911) + (1 minus 120575) 119861 (119911)| minus |119860 (119911) minus (1 + 120575) 119861 (119911)|
ge (2 minus 120575) |119911| minus
infin
sum
119896=2
([119896]119902+ 1 minus 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
|119911|
119896
minus
infin
sum
119896=1
([119896]119902minus 1 + 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
|119911|
119896
minus 120575 |119911| minus
infin
sum
119896=2
([119896]119902minus 1 minus 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
|119911|
119896
minus
infin
sum
119896=1
([119896]119902+ 1 + 120575) Γ (119886
1 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
|119911|
119896
ge 2 (1 minus 120575) |119911| 1 minus
infin
sum
119896=2
[119896]119902minus 120575
1 minus 120575
Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
minus
infin
sum
119896=1
[119896]119902+ 120575
1 minus 120575
Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
= 2 (1 minus 120575) |119911| 1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
minus [
infin
sum
119896=2
(
[119896]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[119896]119902+ 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
)]
times Γ (1198861 119902 119896)
(18)
The last expression is nonnegative by (15) and so 119891 isin
119878
lowast
119867(1198861 120575 119902)
Nowwe obtain the necessary and sufficient conditions for119891 = ℎ + g given by (14)
Theorem 3 Let 119891 = ℎ + 119892 be given by (11) Then 119891 isin
119881
119867(1198861 120575 119902) if and only if
infin
sum
119896=2
(
[119896]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[119896]119902+ 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896)
le 1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
(19)
where 1198861= 1 0 le 120575 lt 1 and Γ(119886
1 119902 119896) is given by (9)
Proof Since 119881
119867(1198861 120575 119902) sub 119878
lowast
119867(1198861 120575 119902) we only to
prove the only if part of the theorem So that for func-tions 119891 isin 119881
119867(1198861 120575 119902) we notice that the condition
(120597120597120579)(arg119867119903119904[1198861 119902]119891(119911)) ge 120575 is equivalent to
120597
120597120579
(arg119867119903119904[1198861 119902] 119891 (119911)) minus 120575
= R119911119863119902(119867
119903
119904[1198861 119902] ℎ (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus
119911119863119902(119867
119903
119904[1198861 119902] 119892 (119911))
119867
119903
119904[1198861 119902] ℎ (119911) + 119867
119903
119904[1198861 119902] 119892 (119911)
minus120575 ge 0
(20)
That is
R[
[
(1 minus 120575) 119911 + sum
infin
119896=2([119896]119902
minus 120575) Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
119911
119896minus sum
infin
119896=1([119896]119902
+ 120575) Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
119911
119896
119911 + sum
infin
119896=2Γ (1198861 119902 119896)
1003816100381610038161003816
119886119896
1003816100381610038161003816
119911
119896+ sum
infin
119896=1Γ (1198861 119902 119896)
1003816100381610038161003816
119887119896
1003816100381610038161003816
119911
119896
]
]
ge 0 (21)
4 ISRNMathematical Analysis
The previous condition must hold for all values of 119911 in UUpon choosing 120601 according to (14) we must have
(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
+ sum
infin
119896=2(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
minus
sum
infin
119896=2(([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
+ sum
infin
119896=2(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
ge 0
(22)
If condition (19) does not hold then the numerator in (22)is negative for 119903 sufficiently close to 1 Hence there exist1199110= 1199030in (01) for which the quotient of (22) is negativeThis
contradicts the fact that 119891 isin 119881119867(1198861 120575 119902) and this completes
the proof
The following theorem gives the distortion bounds forfunctions in119881
119867(1198861 120575 119902)which yield a covering result for this
class
Theorem 4 If 119891 isin 119881119867(1198861 120575 119902) then
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
+
1
Γ (1198861 119902 2)
(
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
) 119903
2
|119911| = 119903 lt 1
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
ge (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
minus
1
Γ (1198861 119902 2)
(
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
) 119903
2
|119911| = 119903 lt 1
(23)
where
Γ (1198861 119902 2) =
(1 minus 1198861) sdot sdot sdot (1 minus 119886
119903)
(1 minus 119902) (1 minus 1198871) sdot sdot sdot (1 minus 119887
119904)
[2]119902= (119902 + 1)
(24)
Proof Wewill only prove the right hand inequalityThe prooffor the left hand inequality is similar
let 119891 isin 119881119867(1198861 120575 119902) Taking the absolute value of 119891 we
obtain
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
infin
sum
119896=2
(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) 119903
119896
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
infin
sum
119896=2
(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) 119903
2
(25)
That is
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1 minus 120575
Γ (1198861 119902 2) ([2]119902
minus 120575)
times
infin
sum
119896=2
(
[2]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[2]119902minus 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
)
times Γ (1198861 119902 2) 119903
2
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1 minus 120575
Γ (1198861 119902 2) ((119902 + 1) minus 120575)
times [1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
] 119903
2
= (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1
Γ (1198861 119902 2)
times [
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
2
(26)
Corollary 5 Let 119891 be of the form (1) so that 119891 isin 119881119867(1198861 120575 119902)
Then
119908 |119908| lt
2Γ (1198861 119902 2) minus 1 minus (Γ (119886
1 119902 2) minus 1) 120575
((119902 + 1) minus 120575) Γ (1198861 119902 2)
minus
2Γ (1198861 119902 2) minus 1 minus (Γ (119886
1 119902 2) minus 1) 120575
((119902 + 1) + 120575) Γ (1198861 119902 2)
1003816100381610038161003816
1198871
1003816100381610038161003816
sub 119891 (U)
(27)
Next one determines the extreme points of closed convex hullof 119881119867(1198861 120575 119902) denoted by clco119881
119867(1198861 120575 119902)
Theorem 6 Set
120582119896=
1 minus 120575
([119896]119902minus 120575) Γ (119886
1 119902 119896)
120583119896=
1 minus 120575
([119896]119902+ 120575) Γ (119886
1 119902 119896)
(28)
For 1198871fixed the extreme points for clco119881
119867(1198861 120575 119902) are
119911 + 120582119896119909119911
119896+ 1198871119911 cup 119911 + 119887
1119911 + 120583119896119909119911
119896 (29)
where 119896 ge 2 and |119909| = 1 minus |1198871|
Proof Any function 119891 isin clco119881119867(1198861 120575 119902) may be expressed
as
119891 (119911) = 119911 +
infin
sum
119896=2
1003816100381610038161003816
119886119896
1003816100381610038161003816
119890
119894120572119896
119911
119896+ 1198871119911 +
infin
sum
119896=2
1003816100381610038161003816
119887119896
1003816100381610038161003816
119890
119894120573119896
119911
119896 (30)
ISRNMathematical Analysis 5
where the coefficients satisfy the inequality (15) Set ℎ1(119911) =
119911 1198921(119911) = 119887
1119911 ℎ119896(119911) = 119911 + 120582
119896119890
119894120572119896
119911
119896 and 119892119896(119911) = 119887
1(119911) +
119890
119894120573119896
119911
119896 for 119896 = 2 3 Writing 119883119896= |119886119896|120582119896 119884119896= |119887119896|120583119896
119896 = 2 3 and1198831= 1minussum
infin
119896=2119883119896 1198841= 1minussum
infin
119896=2119884119896 we have
119891 (119911) =
infin
sum
119896=1
(119883119896ℎ119896 (119911) + 119884119896
119892119896 (119911)) (31)
In particular set
1198911 (119911) = 119911 + 1198871
119911 119891119896 (119911) = 119911 + 120582119896
119909119911
119896+ 1198871119911 + 120583119896119910119911
119896
(119896 ge 2 |119909| +
1003816100381610038161003816
119910
1003816100381610038161003816
= 1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(32)
Therefore the extreme points of clco119881119867(1198861 120575 119902) are
contained in 119891119896(119911) To see that 119891
1is not an extreme point
note that1198911may be written as a convex linear combination of
functions in clco119881119867(1198861 120575 119902) as follows
1198911 (119911) =
1
2
1198911 (119911) + 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
+
1
2
1198911 (119911) minus 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
(33)
If both |119909| = 0 and |119910| = 0 we will show that it can also beexpressed as a convex linear combination of functions inclco119881
119867(1198861 120575 119902) Without loss of generality assume that |119909| ge
|119910| Choose 120598 gt 0 small enough so that 120598 lt |119909||119910| Set119860 = 1 + 120598 and 119861 = 1 minus |120598119909119910| We then see that both
1199051 (119911) = 119911 + 120582119896
119860119909119911
119896+ 1198871119911 + 120583119896119910119861119911
119896
1199052 (119911) = 119911 + 120582119896 (
2 minus 119860) 119909119911
119896+ 1198871119911 + 120583119896119910 (2 minus 119861) 119911
119896
(34)
are in clco119881119867(1198861 120575 119902) and note that
119891119899 (119911) =
1
2
1199051 (119911) + 1199052 (
119911) (35)
The extremal coefficient bound shows that the functions ofthe form (29) are extreme points for clco119881
119867(1198861 120575 119902) and so
the proof is complete
Following the earlier works in [2 13] we refer to the 120574-neighborhood of the functions 119891 defined by (1) to be the setof functions 119865 for which
119873120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
119896 (
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
) +
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(36)
We define the 119902-120574-neighborhood of a function 119891 isin 119878119867as
follows
119873
119902
120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
[119896]119902(
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
)
+
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(37)
In our case let us define the generalized 119902-120574-neighborhood of 119891 to be the set
119873
119902
120574(119891) = 119865
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
]
+ (1 + 120575)
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le (1 minus 120575) 120574
(38)
Theorem 7 Let 119891 be given by (1) If 119891 satisfies the conditions
infin
sum
119896=2
[119896]119902([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
Γ (1198861 119902 119896)
+
infin
sum
1=2
[119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
Γ (1198861 119902 119896) le 1 minus 120575
0 le 120575 lt 1
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(39)
then119873119902120574(119891) sub 119878
lowast
119867(1198861 120575 119902)
Proof Let 119891 satisfy (39) and let
119865 (119911) = 119911 + 1198611119911 +
infin
sum
119896=2
(119860119896119911
119896+ 119861119896119911
119896) (40)
belong to119873119902120574(119891) We have
(1 + 120575)
1003816100381610038161003816
1198611
1003816100381610038161003816
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896
1003816100381610038161003816
]
le (1 + 120575)
1003816100381610038161003816
1198611minus 1198871
1003816100381610038161003816
+ (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
6 ISRNMathematical Analysis
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896minus 119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896minus 119887119896
1003816100381610038161003816
]
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
[2]119902minus 120575
infin
sum
119896=2
Γ (1198861 119902 119896) [[119896]119902
([119896]119902minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ [119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
(119902 + 1) minus 120575
times [(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
] le 1 minus 120575
(41)
Hence
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
119865 isin 119878
lowast
119867(1198861 120575 119902)
(42)
Acknowledgment
The work presented here was partially supported by LRGSTD2011UKMICT0302 and UKM-DLP-2011-050
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A vol 9 pp 3ndash251984
[2] Y Avcı and E Złotkiewicz ldquoOn harmonic univalent mappingsrdquoAnnales Universitatis Mariae Curie-Skłodowska A vol 44 pp1ndash7 1990
[3] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[4] H Exton q-hypergeometric functions and applications EllisHorwood Series Mathematics and its Applications Ellis Hor-wood Chichester UK 1983
[5] H A Ghany ldquo119902-derivative of basic hypergeometric series withrespect to parametersrdquo International Journal of MathematicalAnalysis vol 3 no 33-36 pp 1617ndash1632 2009
[6] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquoMatematicki Vesnik vol 65no 4 pp 454ndash465 2013
[7] J M Jahangiri and H Silverman ldquoHarmonic univalent func-tions with varying argumentsrdquo International Journal of AppliedMathematics vol 8 no 3 pp 267ndash275 2002
[8] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[9] G Murugusundaramoorthy ldquoA class of Ruscheweyh-type har-monic univalent functions with varying argumentsrdquo SouthwestJournal of Pure and AppliedMathematics no 2 pp 90ndash95 2003
[10] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[11] H A Al-Kharsani and R A Al-Khal ldquoUnivalent harmonicfunctionsrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 8 no 2 article 59 p 8 2007
[12] J Dziok and H M Srivastava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
[13] S Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society vol 81 no 4 pp521ndash527 1981
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 ISRNMathematical Analysis
The previous condition must hold for all values of 119911 in UUpon choosing 120601 according to (14) we must have
(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
+ sum
infin
119896=2(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
minus
sum
infin
119896=2(([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
+ sum
infin
119896=2(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) Γ (1198861 119902 119896) 119903
119896minus1
ge 0
(22)
If condition (19) does not hold then the numerator in (22)is negative for 119903 sufficiently close to 1 Hence there exist1199110= 1199030in (01) for which the quotient of (22) is negativeThis
contradicts the fact that 119891 isin 119881119867(1198861 120575 119902) and this completes
the proof
The following theorem gives the distortion bounds forfunctions in119881
119867(1198861 120575 119902)which yield a covering result for this
class
Theorem 4 If 119891 isin 119881119867(1198861 120575 119902) then
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
+
1
Γ (1198861 119902 2)
(
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
) 119903
2
|119911| = 119903 lt 1
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
ge (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
minus
1
Γ (1198861 119902 2)
(
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
) 119903
2
|119911| = 119903 lt 1
(23)
where
Γ (1198861 119902 2) =
(1 minus 1198861) sdot sdot sdot (1 minus 119886
119903)
(1 minus 119902) (1 minus 1198871) sdot sdot sdot (1 minus 119887
119904)
[2]119902= (119902 + 1)
(24)
Proof Wewill only prove the right hand inequalityThe prooffor the left hand inequality is similar
let 119891 isin 119881119867(1198861 120575 119902) Taking the absolute value of 119891 we
obtain
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
infin
sum
119896=2
(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) 119903
119896
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
infin
sum
119896=2
(
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896
1003816100381610038161003816
) 119903
2
(25)
That is
1003816100381610038161003816
119891 (119911)
1003816100381610038161003816
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1 minus 120575
Γ (1198861 119902 2) ([2]119902
minus 120575)
times
infin
sum
119896=2
(
[2]119902minus 120575
1 minus 120575
1003816100381610038161003816
119886119896
1003816100381610038161003816
+
[2]119902minus 120575
1 minus 120575
1003816100381610038161003816
119887119896
1003816100381610038161003816
)
times Γ (1198861 119902 2) 119903
2
le (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1 minus 120575
Γ (1198861 119902 2) ((119902 + 1) minus 120575)
times [1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
] 119903
2
= (1 +
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903 +
1
Γ (1198861 119902 2)
times [
1 minus 120575
(119902 + 1) minus 120575
minus
1 + 120575
(119902 + 1) minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119903
2
(26)
Corollary 5 Let 119891 be of the form (1) so that 119891 isin 119881119867(1198861 120575 119902)
Then
119908 |119908| lt
2Γ (1198861 119902 2) minus 1 minus (Γ (119886
1 119902 2) minus 1) 120575
((119902 + 1) minus 120575) Γ (1198861 119902 2)
minus
2Γ (1198861 119902 2) minus 1 minus (Γ (119886
1 119902 2) minus 1) 120575
((119902 + 1) + 120575) Γ (1198861 119902 2)
1003816100381610038161003816
1198871
1003816100381610038161003816
sub 119891 (U)
(27)
Next one determines the extreme points of closed convex hullof 119881119867(1198861 120575 119902) denoted by clco119881
119867(1198861 120575 119902)
Theorem 6 Set
120582119896=
1 minus 120575
([119896]119902minus 120575) Γ (119886
1 119902 119896)
120583119896=
1 minus 120575
([119896]119902+ 120575) Γ (119886
1 119902 119896)
(28)
For 1198871fixed the extreme points for clco119881
119867(1198861 120575 119902) are
119911 + 120582119896119909119911
119896+ 1198871119911 cup 119911 + 119887
1119911 + 120583119896119909119911
119896 (29)
where 119896 ge 2 and |119909| = 1 minus |1198871|
Proof Any function 119891 isin clco119881119867(1198861 120575 119902) may be expressed
as
119891 (119911) = 119911 +
infin
sum
119896=2
1003816100381610038161003816
119886119896
1003816100381610038161003816
119890
119894120572119896
119911
119896+ 1198871119911 +
infin
sum
119896=2
1003816100381610038161003816
119887119896
1003816100381610038161003816
119890
119894120573119896
119911
119896 (30)
ISRNMathematical Analysis 5
where the coefficients satisfy the inequality (15) Set ℎ1(119911) =
119911 1198921(119911) = 119887
1119911 ℎ119896(119911) = 119911 + 120582
119896119890
119894120572119896
119911
119896 and 119892119896(119911) = 119887
1(119911) +
119890
119894120573119896
119911
119896 for 119896 = 2 3 Writing 119883119896= |119886119896|120582119896 119884119896= |119887119896|120583119896
119896 = 2 3 and1198831= 1minussum
infin
119896=2119883119896 1198841= 1minussum
infin
119896=2119884119896 we have
119891 (119911) =
infin
sum
119896=1
(119883119896ℎ119896 (119911) + 119884119896
119892119896 (119911)) (31)
In particular set
1198911 (119911) = 119911 + 1198871
119911 119891119896 (119911) = 119911 + 120582119896
119909119911
119896+ 1198871119911 + 120583119896119910119911
119896
(119896 ge 2 |119909| +
1003816100381610038161003816
119910
1003816100381610038161003816
= 1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(32)
Therefore the extreme points of clco119881119867(1198861 120575 119902) are
contained in 119891119896(119911) To see that 119891
1is not an extreme point
note that1198911may be written as a convex linear combination of
functions in clco119881119867(1198861 120575 119902) as follows
1198911 (119911) =
1
2
1198911 (119911) + 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
+
1
2
1198911 (119911) minus 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
(33)
If both |119909| = 0 and |119910| = 0 we will show that it can also beexpressed as a convex linear combination of functions inclco119881
119867(1198861 120575 119902) Without loss of generality assume that |119909| ge
|119910| Choose 120598 gt 0 small enough so that 120598 lt |119909||119910| Set119860 = 1 + 120598 and 119861 = 1 minus |120598119909119910| We then see that both
1199051 (119911) = 119911 + 120582119896
119860119909119911
119896+ 1198871119911 + 120583119896119910119861119911
119896
1199052 (119911) = 119911 + 120582119896 (
2 minus 119860) 119909119911
119896+ 1198871119911 + 120583119896119910 (2 minus 119861) 119911
119896
(34)
are in clco119881119867(1198861 120575 119902) and note that
119891119899 (119911) =
1
2
1199051 (119911) + 1199052 (
119911) (35)
The extremal coefficient bound shows that the functions ofthe form (29) are extreme points for clco119881
119867(1198861 120575 119902) and so
the proof is complete
Following the earlier works in [2 13] we refer to the 120574-neighborhood of the functions 119891 defined by (1) to be the setof functions 119865 for which
119873120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
119896 (
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
) +
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(36)
We define the 119902-120574-neighborhood of a function 119891 isin 119878119867as
follows
119873
119902
120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
[119896]119902(
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
)
+
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(37)
In our case let us define the generalized 119902-120574-neighborhood of 119891 to be the set
119873
119902
120574(119891) = 119865
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
]
+ (1 + 120575)
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le (1 minus 120575) 120574
(38)
Theorem 7 Let 119891 be given by (1) If 119891 satisfies the conditions
infin
sum
119896=2
[119896]119902([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
Γ (1198861 119902 119896)
+
infin
sum
1=2
[119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
Γ (1198861 119902 119896) le 1 minus 120575
0 le 120575 lt 1
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(39)
then119873119902120574(119891) sub 119878
lowast
119867(1198861 120575 119902)
Proof Let 119891 satisfy (39) and let
119865 (119911) = 119911 + 1198611119911 +
infin
sum
119896=2
(119860119896119911
119896+ 119861119896119911
119896) (40)
belong to119873119902120574(119891) We have
(1 + 120575)
1003816100381610038161003816
1198611
1003816100381610038161003816
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896
1003816100381610038161003816
]
le (1 + 120575)
1003816100381610038161003816
1198611minus 1198871
1003816100381610038161003816
+ (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
6 ISRNMathematical Analysis
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896minus 119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896minus 119887119896
1003816100381610038161003816
]
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
[2]119902minus 120575
infin
sum
119896=2
Γ (1198861 119902 119896) [[119896]119902
([119896]119902minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ [119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
(119902 + 1) minus 120575
times [(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
] le 1 minus 120575
(41)
Hence
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
119865 isin 119878
lowast
119867(1198861 120575 119902)
(42)
Acknowledgment
The work presented here was partially supported by LRGSTD2011UKMICT0302 and UKM-DLP-2011-050
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A vol 9 pp 3ndash251984
[2] Y Avcı and E Złotkiewicz ldquoOn harmonic univalent mappingsrdquoAnnales Universitatis Mariae Curie-Skłodowska A vol 44 pp1ndash7 1990
[3] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[4] H Exton q-hypergeometric functions and applications EllisHorwood Series Mathematics and its Applications Ellis Hor-wood Chichester UK 1983
[5] H A Ghany ldquo119902-derivative of basic hypergeometric series withrespect to parametersrdquo International Journal of MathematicalAnalysis vol 3 no 33-36 pp 1617ndash1632 2009
[6] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquoMatematicki Vesnik vol 65no 4 pp 454ndash465 2013
[7] J M Jahangiri and H Silverman ldquoHarmonic univalent func-tions with varying argumentsrdquo International Journal of AppliedMathematics vol 8 no 3 pp 267ndash275 2002
[8] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[9] G Murugusundaramoorthy ldquoA class of Ruscheweyh-type har-monic univalent functions with varying argumentsrdquo SouthwestJournal of Pure and AppliedMathematics no 2 pp 90ndash95 2003
[10] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[11] H A Al-Kharsani and R A Al-Khal ldquoUnivalent harmonicfunctionsrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 8 no 2 article 59 p 8 2007
[12] J Dziok and H M Srivastava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
[13] S Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society vol 81 no 4 pp521ndash527 1981
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRNMathematical Analysis 5
where the coefficients satisfy the inequality (15) Set ℎ1(119911) =
119911 1198921(119911) = 119887
1119911 ℎ119896(119911) = 119911 + 120582
119896119890
119894120572119896
119911
119896 and 119892119896(119911) = 119887
1(119911) +
119890
119894120573119896
119911
119896 for 119896 = 2 3 Writing 119883119896= |119886119896|120582119896 119884119896= |119887119896|120583119896
119896 = 2 3 and1198831= 1minussum
infin
119896=2119883119896 1198841= 1minussum
infin
119896=2119884119896 we have
119891 (119911) =
infin
sum
119896=1
(119883119896ℎ119896 (119911) + 119884119896
119892119896 (119911)) (31)
In particular set
1198911 (119911) = 119911 + 1198871
119911 119891119896 (119911) = 119911 + 120582119896
119909119911
119896+ 1198871119911 + 120583119896119910119911
119896
(119896 ge 2 |119909| +
1003816100381610038161003816
119910
1003816100381610038161003816
= 1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(32)
Therefore the extreme points of clco119881119867(1198861 120575 119902) are
contained in 119891119896(119911) To see that 119891
1is not an extreme point
note that1198911may be written as a convex linear combination of
functions in clco119881119867(1198861 120575 119902) as follows
1198911 (119911) =
1
2
1198911 (119911) + 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
+
1
2
1198911 (119911) minus 1205822
(1 minus
1003816100381610038161003816
1198871
1003816100381610038161003816
) 119911
2
(33)
If both |119909| = 0 and |119910| = 0 we will show that it can also beexpressed as a convex linear combination of functions inclco119881
119867(1198861 120575 119902) Without loss of generality assume that |119909| ge
|119910| Choose 120598 gt 0 small enough so that 120598 lt |119909||119910| Set119860 = 1 + 120598 and 119861 = 1 minus |120598119909119910| We then see that both
1199051 (119911) = 119911 + 120582119896
119860119909119911
119896+ 1198871119911 + 120583119896119910119861119911
119896
1199052 (119911) = 119911 + 120582119896 (
2 minus 119860) 119909119911
119896+ 1198871119911 + 120583119896119910 (2 minus 119861) 119911
119896
(34)
are in clco119881119867(1198861 120575 119902) and note that
119891119899 (119911) =
1
2
1199051 (119911) + 1199052 (
119911) (35)
The extremal coefficient bound shows that the functions ofthe form (29) are extreme points for clco119881
119867(1198861 120575 119902) and so
the proof is complete
Following the earlier works in [2 13] we refer to the 120574-neighborhood of the functions 119891 defined by (1) to be the setof functions 119865 for which
119873120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
119896 (
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
) +
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(36)
We define the 119902-120574-neighborhood of a function 119891 isin 119878119867as
follows
119873
119902
120574(119891) = 119865 = 119911 +
infin
sum
119896=2
119860119896119911
119896+
infin
sum
119896=1
119861119896119911
119896
infin
sum
119896=2
[119896]119902(
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
)
+
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le 120574
(37)
In our case let us define the generalized 119902-120574-neighborhood of 119891 to be the set
119873
119902
120574(119891) = 119865
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896minus 119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896minus 119861119896
1003816100381610038161003816
]
+ (1 + 120575)
1003816100381610038161003816
1198871minus 1198611
1003816100381610038161003816
le (1 minus 120575) 120574
(38)
Theorem 7 Let 119891 be given by (1) If 119891 satisfies the conditions
infin
sum
119896=2
[119896]119902([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
Γ (1198861 119902 119896)
+
infin
sum
1=2
[119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
Γ (1198861 119902 119896) le 1 minus 120575
0 le 120575 lt 1
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
(39)
then119873119902120574(119891) sub 119878
lowast
119867(1198861 120575 119902)
Proof Let 119891 satisfy (39) and let
119865 (119911) = 119911 + 1198611119911 +
infin
sum
119896=2
(119860119896119911
119896+ 119861119896119911
119896) (40)
belong to119873119902120574(119891) We have
(1 + 120575)
1003816100381610038161003816
1198611
1003816100381610038161003816
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896
1003816100381610038161003816
]
le (1 + 120575)
1003816100381610038161003816
1198611minus 1198871
1003816100381610038161003816
+ (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
6 ISRNMathematical Analysis
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896minus 119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896minus 119887119896
1003816100381610038161003816
]
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
[2]119902minus 120575
infin
sum
119896=2
Γ (1198861 119902 119896) [[119896]119902
([119896]119902minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ [119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
(119902 + 1) minus 120575
times [(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
] le 1 minus 120575
(41)
Hence
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
119865 isin 119878
lowast
119867(1198861 120575 119902)
(42)
Acknowledgment
The work presented here was partially supported by LRGSTD2011UKMICT0302 and UKM-DLP-2011-050
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A vol 9 pp 3ndash251984
[2] Y Avcı and E Złotkiewicz ldquoOn harmonic univalent mappingsrdquoAnnales Universitatis Mariae Curie-Skłodowska A vol 44 pp1ndash7 1990
[3] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[4] H Exton q-hypergeometric functions and applications EllisHorwood Series Mathematics and its Applications Ellis Hor-wood Chichester UK 1983
[5] H A Ghany ldquo119902-derivative of basic hypergeometric series withrespect to parametersrdquo International Journal of MathematicalAnalysis vol 3 no 33-36 pp 1617ndash1632 2009
[6] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquoMatematicki Vesnik vol 65no 4 pp 454ndash465 2013
[7] J M Jahangiri and H Silverman ldquoHarmonic univalent func-tions with varying argumentsrdquo International Journal of AppliedMathematics vol 8 no 3 pp 267ndash275 2002
[8] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[9] G Murugusundaramoorthy ldquoA class of Ruscheweyh-type har-monic univalent functions with varying argumentsrdquo SouthwestJournal of Pure and AppliedMathematics no 2 pp 90ndash95 2003
[10] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[11] H A Al-Kharsani and R A Al-Khal ldquoUnivalent harmonicfunctionsrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 8 no 2 article 59 p 8 2007
[12] J Dziok and H M Srivastava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
[13] S Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society vol 81 no 4 pp521ndash527 1981
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 ISRNMathematical Analysis
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119860119896minus 119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119861119896minus 119887119896
1003816100381610038161003816
]
+
infin
sum
119896=2
Γ (1198861 119902 119896) [([119896]119902
minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ ([119896]119902+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
[2]119902minus 120575
infin
sum
119896=2
Γ (1198861 119902 119896) [[119896]119902
([119896]119902minus 120575)
1003816100381610038161003816
119886119896
1003816100381610038161003816
+ [119896]119902([119896]119902
+ 120575)
1003816100381610038161003816
119887119896
1003816100381610038161003816
]
le (1 minus 120575) 120574 + (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
+
1
(119902 + 1) minus 120575
times [(1 minus 120575) minus (1 + 120575)
1003816100381610038161003816
1198871
1003816100381610038161003816
] le 1 minus 120575
(41)
Hence
120574 le
1 minus 120575
(119902 + 1) minus 120575
(1 minus
1 + 120575
1 minus 120575
1003816100381610038161003816
1198871
1003816100381610038161003816
)
119865 isin 119878
lowast
119867(1198861 120575 119902)
(42)
Acknowledgment
The work presented here was partially supported by LRGSTD2011UKMICT0302 and UKM-DLP-2011-050
References
[1] J Clunie and T Sheil-Small ldquoHarmonic univalent functionsrdquoAnnales Academiae Scientiarum Fennicae A vol 9 pp 3ndash251984
[2] Y Avcı and E Złotkiewicz ldquoOn harmonic univalent mappingsrdquoAnnales Universitatis Mariae Curie-Skłodowska A vol 44 pp1ndash7 1990
[3] G Gasper and M Rahman Basic Hypergeometric Series vol 35of Encyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 1990
[4] H Exton q-hypergeometric functions and applications EllisHorwood Series Mathematics and its Applications Ellis Hor-wood Chichester UK 1983
[5] H A Ghany ldquo119902-derivative of basic hypergeometric series withrespect to parametersrdquo International Journal of MathematicalAnalysis vol 3 no 33-36 pp 1617ndash1632 2009
[6] A Mohammed and M Darus ldquoA generalized operator involv-ing the 119902-hypergeometric functionrdquoMatematicki Vesnik vol 65no 4 pp 454ndash465 2013
[7] J M Jahangiri and H Silverman ldquoHarmonic univalent func-tions with varying argumentsrdquo International Journal of AppliedMathematics vol 8 no 3 pp 267ndash275 2002
[8] J M Jahangiri ldquoHarmonic functions starlike in the unit diskrdquoJournal of Mathematical Analysis and Applications vol 235 no2 pp 470ndash477 1999
[9] G Murugusundaramoorthy ldquoA class of Ruscheweyh-type har-monic univalent functions with varying argumentsrdquo SouthwestJournal of Pure and AppliedMathematics no 2 pp 90ndash95 2003
[10] S Ruscheweyh ldquoNew criteria for univalent functionsrdquo Proceed-ings of the American Mathematical Society vol 49 pp 109ndash1151975
[11] H A Al-Kharsani and R A Al-Khal ldquoUnivalent harmonicfunctionsrdquo Journal of Inequalities in Pure and Applied Mathe-matics vol 8 no 2 article 59 p 8 2007
[12] J Dziok and H M Srivastava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
[13] S Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society vol 81 no 4 pp521ndash527 1981
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
