Embed Size (px)
Citation preview
Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 958796 6 pageshttpdxdoiorg1011552013958796
Research ArticleCertain Subclasses of Analytic Functions with Complex Order
A Selvam1 P Sooriya Kala1 and N Marikkannan2
1 Department of Mathematics VHNSN College Virudhunagar 626001 India2Department of Mathematics Government Arts College Melur 625106 India
Correspondence should be addressed to N Marikkannan natarajanmarikkannangmailcom
Received 12 August 2013 Accepted 12 September 2013
Academic Editors G Bonanno and A Ibeas
Copyright copy 2013 A Selvam et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Two new subclasses of analytic functions of complex order are introduced Apart from establishing coefficient bounds for theseclasses we establish inclusion relationships involving (119899-120575) neighborhoods of analytic functionswith negative coefficients belongingto these subclasses
1 Introduction
LetA(119899) denote the class of functions of the form
119891 (119911) = 119911 minus
infin
sum
119896=119899+1
119886119896119911119896
(119886119896ge 0 119899 isin N = 1 2 3 ) (1)
which are analytic and univalent in the open disc
Δ = 119911 isin C |119911| lt 1 (2)
A function 119891(119911) isin A(119899) is star-like of complex order 119887denoted as 119891(119911) isin 119878
lowast
(119887) if and only if it satisfies
R1 +
1
119887
(
1199111198911015840
119891
minus 1) gt 0 (119911 isin Δ) (3)
A function 119891(119911) isin A(119899) is convex of complex order 119887denoted as 119891(119911) isin 119862(119887) if and only if it satisfies
R1 +
1
119887
(
11991111989110158401015840
1198911015840) gt 0 (119911 isin Δ) (4)
These classes 119878lowast(119887) and 119862(119887) are introduced and studied byNasr and Aouf [1] and Wiatrowski [2]
For the two functions 119891119895(119895 = 1 2) given by
119891119895(119911) = 119911 +
infin
sum
119896=2
119886119896119895
119911119896
(119895 = 1 2) (5)
the Hadamard product or convolution denoted by (1198911
lowast
1198912)(119911) is given by
(1198911lowast 1198912) (119911) = 119911 +
infin
sum
119896=2
1198861198961
1198861198962
119911119896
(6)
Given 119891(119911) of the form (1) and 120575 ge 0 we define 119899-120575neighborhood of a function 119891 isin A(119899) as
119873119899120575
(119891) = 119892 isin A (119899) |
infin
sum
119896=119899+1
1198961003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le 120575 (7)
In particular for the identity function 119890(119911) = 119911
119873119899120575
(119890) = 119892 isin A (119899) |
infin
sum
119896=119899+1
1198961003816100381610038161003816119887119896
1003816100381610038161003816le 120575 (8)
The concept of Neighborhood 119873119899120575
of a function 119891 isintroduced and studied by Ruscheweyh [3] and extendedfurther by Silverman [4]
For complex numbers 1205721 1205722 120572
119902and 120573
1 1205732 120573
119904
(120573119895
isin C Zminus0 Zminus0
= 0 minus1 minus2 for 119895 = 1 2 119904)
2 The Scientific World Journal
we define the generalized hypergeometric function119902119865119904(1205721 1205722 120572
119902 1205731 1205732 120573
119904 119911) as
119902119865119904(1205721 1205722 120572
119902 1205731 1205732 120573
119904 119911)
=
infin
sum
119896=0
(1205721)119896(1205722)119896sdot sdot sdot (120572119902)119896
119911119896
(1205731)119896(1205732)119896sdot sdot sdot (120573119904)119896119896
(119902 le 119904 + 1
119902 119904 isin N0= N cup 0 119911 isin 119880)
(9)
whereN denotes the set of all positive integers and (119909)119896is the
Pochhammer symbol defined in terms of gamma functionsas
(119909)119896=
Γ(119909 + 119896)
Γ(119909)
=
1 if 119896 = 0
119909 (119909 + 1) sdot sdot sdot (119909 + 119896 minus 1) if 119896 isin N
(10)
Corresponding to the function 119892119902119904
(1205721 1205731 119911) defined by
119892119902119904
(1205721 1205731 119911)
= 119911119902119865119904(1205721 1205722 120572
119902 1205731 1205732 120573
119904 119911)
(11)
recently in [5] an operator D119898120582120583
(1205721 1205731)119891(119911) A rarr A is
defined by
D0
120582120583(1205721 1205731) 119891(119911) = 119891(119911) lowast 119892
119902119904(1205721 1205731 119911)
D1
120582120583(1205721 1205731) 119891(119911) = (1 minus 120582 + 120583) (119891(119911) lowast 119892
119902119904(1205721 1205731 119911))
+ (120582 minus 120583) 119911(119891(119911) lowast 119892119902119904
(1205721 1205731 119911))
1015840
+ 1205821205831199112
(119891(119911) lowast 119892119902119904
(1205721 1205731 119911))
10158401015840
D119898
120582120583(1205721 1205731) 119891(119911) = D
1
120582120583(D119898minus1
120582120583(1205721 1205731) 119891(119911))
(12)
where 0 le 120583 le 120582 le 1 and 119898 isin N0 By the above definition it
is easy to note that
D119898
120582120583(1205721 1205731) 119891(119911)
= 119911 +
infin
sum
119896=2
[1 + (119896 minus 1) (120582 minus 120583 + 119896120583120582)]119898
times
(1205721)119896minus1
(1205722)119896minus1
(120572119902)119896minus1
(1205731)119896minus1
(1205732)119896minus1
(120573119904)119896minus1
(119896 minus 1)
119886119896119911119896
(13)
Let us take for convenience that
119861119896=
(1205721)119896minus1
(1205722)119896minus1
(120572119902)119896minus1
(1205731)119896minus1
(1205732)119896minus1
(120573119904)119896minus1
(119896 minus 1)
119862119896= 1 + (119896 minus 1) (120582 minus 120583 + 119896120583120582)
(14)
Hence we have
D119898
120582120583(1205721 1205731) 119891(119911) = 119911 +
infin
sum
119896=2
119862119898
119896119861119896119886119896119911119896
(15)
For suitable values of 1205721198941015840119904 1205731198951015840119904 119902 119904 119898 120582 and 120583 we can
deduce several operators such as Salagean derivative operator[6] Ruscheweyh derivative operator [7] fractional calculusoperator [8] Carlson-Shaffer operator [9] Dziok-Srivatsavaoperator [10] and also the operator introduced by Abubakerand Darus [11]
Definition 1 For 0 le 120572 le 1 we let 119860 be the subclass ofA(119899)
consisting of functions of the form (1) that satisfy
10038161003816100381610038161003816100381610038161003816
1
119887
(119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
times ( (1 minus 120572)D119898
120582120583(1205721 1205731) 119891 (119911)
+ 120572119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
)
minus1
)
10038161003816100381610038161003816100381610038161003816
lt 120574
(16)
where 119911 isin Δ 119887 isin C 0 0 lt 120574 le 1 andD119898120582120583
(1205721 1205731)119891(119911) are
as given in (15)
Definition 2 For 0 le 120572 le 1 we let 119861 be the subclass ofA(119899)consisting of functions of the form (1) that satisfy
1003816100381610038161003816100381610038161003816100381610038161003816
1
119887
((1 minus 120572)
D119898120582120583
(1205721 1205731) 119891 (119911)
119911
+ 120572[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
minus 1)
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120574
(17)
where 119911 isin Δ 119887 isin C 0 0 lt 120574 le 1 andD119898120582120583
(1205721 1205731)119891(119911) are
as given in (15)
By specializing the parameters involved in the abovedefinitions we could arrive at several known as well as newclasses For example by taking 120582 = 1 120583 = 0 119902 = 2 119904 = 11205721= 1205731 and 120572
2= 1 and the above classes reduced to
1198601= 119891 isin A (119899) |
1003816100381610038161003816100381610038161003816
1
119887
(119911[D119898
119891 (119911)]1015840
times ( (1 minus 120572)D119898
119891 (119911)
+ 120572119911[D119898
119891 (119911)]1015840
)
minus1
)
1003816100381610038161003816100381610038161003816
lt 120574
(18)
The Scientific World Journal 3
where D119898119891(119911) denote the Salagean derivative of order 119898
given by
D119898
119891 (119911) = 119911 minus
infin
sum
119896=119899+1
119896119898
119886119896119911119896
1198611= 119891 isin A (119899) |
10038161003816100381610038161003816100381610038161003816
1
119887
((1 minus 120572)
D119898119891 (119911)
119911
+ 120572[D119898
119891 (119911)]1015840
minus 1)
10038161003816100381610038161003816100381610038161003816
lt 120574
(19)
Similarly on taking 119902 = 2 119904 = 1 1205721
= 120578 minus 1 (120578 gt minus1)1205722= 1 120573
1= 1 one gets
1198602= 119891 isin A (119899) |
1003816100381610038161003816100381610038161003816
1
119887
(119911[119863119898
119891 (119911)]1015840
times ((1 minus 120572)119863119898
119891 (119911)
+ 120572119911[119863119898
119891 (119911)]1015840
)
minus1
)
1003816100381610038161003816100381610038161003816
lt 120574
(20)
where 119863119898
119891(119911) is the operator introduced and studied byAbubaker and Darus [11] given by
119863119898
119891 (119911) = 119911 minus
infin
sum
119896=119899+1
119862119898
119896(
120578 + 119896 minus 1
119896 minus 1) 119911119896
1198612=119891 isin A (119899) |
10038161003816100381610038161003816100381610038161003816
1
119887
((1 minus 120572)
119863119898
119891 (119911)
119911
+ 120572[119863119898
119891 (119911)]1015840
minus 1)
10038161003816100381610038161003816100381610038161003816
lt 120574
(21)
Further by taking 119898 = 0 in the definition of the classes119860 and 119861 we could arrive at 119878
119899(119902 119904 120572 119887 120574) and 119877
119899(119902 119904 120572 119887 120574)
whichwere introduced and studied byMurugusundaramoor-thy et al [12]
In this paper we establish the coefficient inequalitiesfor the classes 119860 and 119861 and several inclusion relationshipsinvolving 119899-120575 neighborhoods of analytic univalent functionswith negative and missing coefficients belonging to theseclasses
2 Coefficient Inequalities
Theorem 3 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860 if and only ifinfin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
119896119861119896119886119896le 120574 |119887| (22)
Proof Let the functions of form (1) belong to the class 119860Then in view of (15) and (16) we get
1003816100381610038161003816100381610038161003816100381610038161003816
suminfin
119896=119899+1([1 + 120572 (119896 minus 1)] minus 119896) 119862
119898
119896119861119896119886119896119911119896
119911 minus suminfin
119896=119899+1[1 + 120572 (119896 minus 1)] 119862
119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120574 |119887|
suminfin
119896=119899+1([1 + 120572 (119896 minus 1)] minus 119896) 119862
119898
119896119861119896119886119896119903119896minus1
1 minus suminfin
119896=119899+1[1 + 120572 (119896 minus 1)] 119862
119898
119896119861119896119886119896119903119896minus1
lt 120574 |119887|
(23)
Letting 119903 rarr 1minus through real values we get the required
assertion (22) Conversely suppose 119891(119911) satisfies (22) thenin view of (16) consider
100381610038161003816100381610038161003816
119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
minus (1 minus 120572)D119898
120582120583(1205721 1205731) 119891 (119911)
minus120572119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840100381610038161003816100381610038161003816
minus 120574 |119887|
100381610038161003816100381610038161003816
(1 minus 120572)D119898
120582120583(1205721 1205731) 119891 (119911)
+ 120572119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840100381610038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816100381610038161003816
infin
sum
119896=119899+1
([1 + 120572 (119896 minus 1)] minus 119896) 119862119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
minus 120574 |119887|
1003816100381610038161003816100381610038161003816100381610038161003816
119911 minus
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
lt
infin
sum
119896=119899+1
([1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896)
times 119862119898
119896119861119896119886119896minus 120574 |119887|
le 0
(24)
Hence the result follows
Similarly we prove the following
Theorem 4 Let the function 119891 isin A(119899) be as defined in (1)Then 119891 isin 119861 if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896119861119896119886119896le 120574 |119887| (25)
Corollary 5 Let the function 119891 isin A(119899) as given in (1) Then119891(119911) isin 119860
1if and only if
infin
sum
119896=119899+1
[1+120572 (119896minus1)] (120574 |119887|minus1)+119896 119896119898
119886119896le120574 |119887| (26)
Corollary 6 Let the function 119891 isin A(119899) be as defined in (1)Then 119891 isin 119861
1if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119896119898
119886119896le 120574 |119887| (27)
Corollary 7 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860
2if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
119896
times (
120578 + 119896 minus 1
119896 minus 1) 119886119896le 120574 |119887|
(28)
4 The Scientific World Journal
Corollary 8 Let the function 119891(119911) isin A(119899) be as defined in(1) Then 119891 isin 119861
2if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896(
120578 + 119896 minus 1
119896 minus 1) 119886119896le 120574 |119887| (29)
3 Inclusion Relationships
Theorem 9 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(120574 |119887| gt 1)
(30)
then 119860 sub 119873119899120575
(119890)
Proof Let 119891 isin 119860 Then in view of (22) we have
119862119898
119899+1119861119899+1
[(1+119899120572) (120574 |119887|minus1) 119899+1]
infin
sum
119896=119899+1
119886119896
le 120574 |119887|
(31)
infin
sum
119896=119899+1
119886119896le
120574 |119887|
119862119898
119899+1119861119899+1
[(1 + 119899120572) (120574 |119887| minus 1) 119899 + 1]
(32)
Consider
119862119898
119899+1119861119899+1
infin
sum
119896=119899+1
119896119886119896
le 120574 |119887| + (1 + 119899120572) (1 minus 120574 |119887|) 119861119899+1
119862119898
119899+1
infin
sum
119896=119899+1
119886119896
le 120574 |119887| +
(1 + 119899120572) (1 minus 120574 |119887|) 120574 |119887|
(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1
le
120574 |119887| (119899 + 1)
(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1
(33)
Hence
infin
sum
119896=119899+1
119896119886119896le
120574 |119887| (119899 + 1)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
= 120575
(34)
Hence the result follows
In similar manner we establish the following result
Theorem 10 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(35)
then 119861 sub 119873119899120575
(119890)
Corollary 11 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898 (36)
then 1198601sub 119873119899120575
(119890)
Corollary 12 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) (119899 + 1)119898 (37)
then 1198611sub 119873119899120575
(119890)
Corollary 13 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119862119898
119899+1(120578+119899
119899)
(38)
then 1198602sub 119873119899120575
(119890)
Corollary 14 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) 119862119898
119899+1(120578+119899
119899)
(39)
then 1198612sub 119873119899120575
(119890)
4 Neighborhoods for 119860120590 and 119861
120590
In this section we determine the neighborhood properties of119860120590 and 119861
120590 Here the classes119860120590 consist of functions119891 isin A(119899)
for which there exists a function 119892(119911) isin 119860 such that
10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt 1 minus 120590 (119911 isin Δ 0 le 120590 lt 1) (40)
In the same way we define 119861120590 consisting of functions 119891(119911) isin
A(119899) for which there exists another function 119892(119911) isin 119861 suchthat
10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt 1 minus 120590 (119911 isin Δ 0 le 120590 lt 1) (41)
Theorem 15 If 119892 isin 119860 and
120590 = 1 minus
120575
119899 + 1
times [(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times 119861119899+1
119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times119861119899+1
119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(42)
then 119873119899120575
(119892) sub 119860120590
The Scientific World Journal 5
Proof Suppose 119891 isin 119873119899120575
(119892) then
infin
sum
119896=119899+1
1198961003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le 120575
infin
sum
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le
120575
119899 + 1
(43)
Since 119892 isin 119860 we have
infin
sum
119896=119899+1
119887119896le
119887 |119896|
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(44)
Consider10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt
suminfin
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816
1 minus suminfin
119896=119899+1119887119896
le
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1minus 120574 |119887|
= 1 minus 120590
(45)
Therefore 119891 isin 119860120590 for 120590 given by (42)
Theorem 16 If 119892 isin 119861 and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(1 + 119899120572) 119861119899+1
119862119898
119899+1minus 120574 |119887|
(46)
then 119873119899120575
(119892) sub 119861120590
Corollary 17 If 119892 isin 1198601and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (119899 + 1)119898
minus 120574 |119887|)minus1
(120574 |119887| gt 1)
(47)
then 119873119899120575
(119892) sub 119860120590
1
Corollary 18 If 119892 isin 1198611and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (119899 + 1)119898
(1 + 119899120572) (119899 + 1)119898
minus 120574 |119887|
(48)
then 119873119899120575
(119892) sub 119861120590
1
Corollary 19 If 119892 isin 1198602and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (
120578 + 119899
119899)119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
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
2 The Scientific World Journal
we define the generalized hypergeometric function119902119865119904(1205721 1205722 120572
119902 1205731 1205732 120573
119904 119911) as
119902119865119904(1205721 1205722 120572
119902 1205731 1205732 120573
119904 119911)
=
infin
sum
119896=0
(1205721)119896(1205722)119896sdot sdot sdot (120572119902)119896
119911119896
(1205731)119896(1205732)119896sdot sdot sdot (120573119904)119896119896
(119902 le 119904 + 1
119902 119904 isin N0= N cup 0 119911 isin 119880)
(9)
whereN denotes the set of all positive integers and (119909)119896is the
Pochhammer symbol defined in terms of gamma functionsas
(119909)119896=
Γ(119909 + 119896)
Γ(119909)
=
1 if 119896 = 0
119909 (119909 + 1) sdot sdot sdot (119909 + 119896 minus 1) if 119896 isin N
(10)
Corresponding to the function 119892119902119904
(1205721 1205731 119911) defined by
119892119902119904
(1205721 1205731 119911)
= 119911119902119865119904(1205721 1205722 120572
119902 1205731 1205732 120573
119904 119911)
(11)
recently in [5] an operator D119898120582120583
(1205721 1205731)119891(119911) A rarr A is
defined by
D0
120582120583(1205721 1205731) 119891(119911) = 119891(119911) lowast 119892
119902119904(1205721 1205731 119911)
D1
120582120583(1205721 1205731) 119891(119911) = (1 minus 120582 + 120583) (119891(119911) lowast 119892
119902119904(1205721 1205731 119911))
+ (120582 minus 120583) 119911(119891(119911) lowast 119892119902119904
(1205721 1205731 119911))
1015840
+ 1205821205831199112
(119891(119911) lowast 119892119902119904
(1205721 1205731 119911))
10158401015840
D119898
120582120583(1205721 1205731) 119891(119911) = D
1
120582120583(D119898minus1
120582120583(1205721 1205731) 119891(119911))
(12)
where 0 le 120583 le 120582 le 1 and 119898 isin N0 By the above definition it
is easy to note that
D119898
120582120583(1205721 1205731) 119891(119911)
= 119911 +
infin
sum
119896=2
[1 + (119896 minus 1) (120582 minus 120583 + 119896120583120582)]119898
times
(1205721)119896minus1
(1205722)119896minus1
(120572119902)119896minus1
(1205731)119896minus1
(1205732)119896minus1
(120573119904)119896minus1
(119896 minus 1)
119886119896119911119896
(13)
Let us take for convenience that
119861119896=
(1205721)119896minus1
(1205722)119896minus1
(120572119902)119896minus1
(1205731)119896minus1
(1205732)119896minus1
(120573119904)119896minus1
(119896 minus 1)
119862119896= 1 + (119896 minus 1) (120582 minus 120583 + 119896120583120582)
(14)
Hence we have
D119898
120582120583(1205721 1205731) 119891(119911) = 119911 +
infin
sum
119896=2
119862119898
119896119861119896119886119896119911119896
(15)
For suitable values of 1205721198941015840119904 1205731198951015840119904 119902 119904 119898 120582 and 120583 we can
deduce several operators such as Salagean derivative operator[6] Ruscheweyh derivative operator [7] fractional calculusoperator [8] Carlson-Shaffer operator [9] Dziok-Srivatsavaoperator [10] and also the operator introduced by Abubakerand Darus [11]
Definition 1 For 0 le 120572 le 1 we let 119860 be the subclass ofA(119899)
consisting of functions of the form (1) that satisfy
10038161003816100381610038161003816100381610038161003816
1
119887
(119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
times ( (1 minus 120572)D119898
120582120583(1205721 1205731) 119891 (119911)
+ 120572119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
)
minus1
)
10038161003816100381610038161003816100381610038161003816
lt 120574
(16)
where 119911 isin Δ 119887 isin C 0 0 lt 120574 le 1 andD119898120582120583
(1205721 1205731)119891(119911) are
as given in (15)
Definition 2 For 0 le 120572 le 1 we let 119861 be the subclass ofA(119899)consisting of functions of the form (1) that satisfy
1003816100381610038161003816100381610038161003816100381610038161003816
1
119887
((1 minus 120572)
D119898120582120583
(1205721 1205731) 119891 (119911)
119911
+ 120572[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
minus 1)
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120574
(17)
where 119911 isin Δ 119887 isin C 0 0 lt 120574 le 1 andD119898120582120583
(1205721 1205731)119891(119911) are
as given in (15)
By specializing the parameters involved in the abovedefinitions we could arrive at several known as well as newclasses For example by taking 120582 = 1 120583 = 0 119902 = 2 119904 = 11205721= 1205731 and 120572
2= 1 and the above classes reduced to
1198601= 119891 isin A (119899) |
1003816100381610038161003816100381610038161003816
1
119887
(119911[D119898
119891 (119911)]1015840
times ( (1 minus 120572)D119898
119891 (119911)
+ 120572119911[D119898
119891 (119911)]1015840
)
minus1
)
1003816100381610038161003816100381610038161003816
lt 120574
(18)
The Scientific World Journal 3
where D119898119891(119911) denote the Salagean derivative of order 119898
given by
D119898
119891 (119911) = 119911 minus
infin
sum
119896=119899+1
119896119898
119886119896119911119896
1198611= 119891 isin A (119899) |
10038161003816100381610038161003816100381610038161003816
1
119887
((1 minus 120572)
D119898119891 (119911)
119911
+ 120572[D119898
119891 (119911)]1015840
minus 1)
10038161003816100381610038161003816100381610038161003816
lt 120574
(19)
Similarly on taking 119902 = 2 119904 = 1 1205721
= 120578 minus 1 (120578 gt minus1)1205722= 1 120573
1= 1 one gets
1198602= 119891 isin A (119899) |
1003816100381610038161003816100381610038161003816
1
119887
(119911[119863119898
119891 (119911)]1015840
times ((1 minus 120572)119863119898
119891 (119911)
+ 120572119911[119863119898
119891 (119911)]1015840
)
minus1
)
1003816100381610038161003816100381610038161003816
lt 120574
(20)
where 119863119898
119891(119911) is the operator introduced and studied byAbubaker and Darus [11] given by
119863119898
119891 (119911) = 119911 minus
infin
sum
119896=119899+1
119862119898
119896(
120578 + 119896 minus 1
119896 minus 1) 119911119896
1198612=119891 isin A (119899) |
10038161003816100381610038161003816100381610038161003816
1
119887
((1 minus 120572)
119863119898
119891 (119911)
119911
+ 120572[119863119898
119891 (119911)]1015840
minus 1)
10038161003816100381610038161003816100381610038161003816
lt 120574
(21)
Further by taking 119898 = 0 in the definition of the classes119860 and 119861 we could arrive at 119878
119899(119902 119904 120572 119887 120574) and 119877
119899(119902 119904 120572 119887 120574)
whichwere introduced and studied byMurugusundaramoor-thy et al [12]
In this paper we establish the coefficient inequalitiesfor the classes 119860 and 119861 and several inclusion relationshipsinvolving 119899-120575 neighborhoods of analytic univalent functionswith negative and missing coefficients belonging to theseclasses
2 Coefficient Inequalities
Theorem 3 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860 if and only ifinfin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
119896119861119896119886119896le 120574 |119887| (22)
Proof Let the functions of form (1) belong to the class 119860Then in view of (15) and (16) we get
1003816100381610038161003816100381610038161003816100381610038161003816
suminfin
119896=119899+1([1 + 120572 (119896 minus 1)] minus 119896) 119862
119898
119896119861119896119886119896119911119896
119911 minus suminfin
119896=119899+1[1 + 120572 (119896 minus 1)] 119862
119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120574 |119887|
suminfin
119896=119899+1([1 + 120572 (119896 minus 1)] minus 119896) 119862
119898
119896119861119896119886119896119903119896minus1
1 minus suminfin
119896=119899+1[1 + 120572 (119896 minus 1)] 119862
119898
119896119861119896119886119896119903119896minus1
lt 120574 |119887|
(23)
Letting 119903 rarr 1minus through real values we get the required
assertion (22) Conversely suppose 119891(119911) satisfies (22) thenin view of (16) consider
100381610038161003816100381610038161003816
119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
minus (1 minus 120572)D119898
120582120583(1205721 1205731) 119891 (119911)
minus120572119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840100381610038161003816100381610038161003816
minus 120574 |119887|
100381610038161003816100381610038161003816
(1 minus 120572)D119898
120582120583(1205721 1205731) 119891 (119911)
+ 120572119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840100381610038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816100381610038161003816
infin
sum
119896=119899+1
([1 + 120572 (119896 minus 1)] minus 119896) 119862119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
minus 120574 |119887|
1003816100381610038161003816100381610038161003816100381610038161003816
119911 minus
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
lt
infin
sum
119896=119899+1
([1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896)
times 119862119898
119896119861119896119886119896minus 120574 |119887|
le 0
(24)
Hence the result follows
Similarly we prove the following
Theorem 4 Let the function 119891 isin A(119899) be as defined in (1)Then 119891 isin 119861 if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896119861119896119886119896le 120574 |119887| (25)
Corollary 5 Let the function 119891 isin A(119899) as given in (1) Then119891(119911) isin 119860
1if and only if
infin
sum
119896=119899+1
[1+120572 (119896minus1)] (120574 |119887|minus1)+119896 119896119898
119886119896le120574 |119887| (26)
Corollary 6 Let the function 119891 isin A(119899) be as defined in (1)Then 119891 isin 119861
1if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119896119898
119886119896le 120574 |119887| (27)
Corollary 7 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860
2if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
119896
times (
120578 + 119896 minus 1
119896 minus 1) 119886119896le 120574 |119887|
(28)
4 The Scientific World Journal
Corollary 8 Let the function 119891(119911) isin A(119899) be as defined in(1) Then 119891 isin 119861
2if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896(
120578 + 119896 minus 1
119896 minus 1) 119886119896le 120574 |119887| (29)
3 Inclusion Relationships
Theorem 9 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(120574 |119887| gt 1)
(30)
then 119860 sub 119873119899120575
(119890)
Proof Let 119891 isin 119860 Then in view of (22) we have
119862119898
119899+1119861119899+1
[(1+119899120572) (120574 |119887|minus1) 119899+1]
infin
sum
119896=119899+1
119886119896
le 120574 |119887|
(31)
infin
sum
119896=119899+1
119886119896le
120574 |119887|
119862119898
119899+1119861119899+1
[(1 + 119899120572) (120574 |119887| minus 1) 119899 + 1]
(32)
Consider
119862119898
119899+1119861119899+1
infin
sum
119896=119899+1
119896119886119896
le 120574 |119887| + (1 + 119899120572) (1 minus 120574 |119887|) 119861119899+1
119862119898
119899+1
infin
sum
119896=119899+1
119886119896
le 120574 |119887| +
(1 + 119899120572) (1 minus 120574 |119887|) 120574 |119887|
(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1
le
120574 |119887| (119899 + 1)
(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1
(33)
Hence
infin
sum
119896=119899+1
119896119886119896le
120574 |119887| (119899 + 1)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
= 120575
(34)
Hence the result follows
In similar manner we establish the following result
Theorem 10 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(35)
then 119861 sub 119873119899120575
(119890)
Corollary 11 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898 (36)
then 1198601sub 119873119899120575
(119890)
Corollary 12 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) (119899 + 1)119898 (37)
then 1198611sub 119873119899120575
(119890)
Corollary 13 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119862119898
119899+1(120578+119899
119899)
(38)
then 1198602sub 119873119899120575
(119890)
Corollary 14 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) 119862119898
119899+1(120578+119899
119899)
(39)
then 1198612sub 119873119899120575
(119890)
4 Neighborhoods for 119860120590 and 119861
120590
In this section we determine the neighborhood properties of119860120590 and 119861
120590 Here the classes119860120590 consist of functions119891 isin A(119899)
for which there exists a function 119892(119911) isin 119860 such that
10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt 1 minus 120590 (119911 isin Δ 0 le 120590 lt 1) (40)
In the same way we define 119861120590 consisting of functions 119891(119911) isin
A(119899) for which there exists another function 119892(119911) isin 119861 suchthat
10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt 1 minus 120590 (119911 isin Δ 0 le 120590 lt 1) (41)
Theorem 15 If 119892 isin 119860 and
120590 = 1 minus
120575
119899 + 1
times [(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times 119861119899+1
119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times119861119899+1
119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(42)
then 119873119899120575
(119892) sub 119860120590
The Scientific World Journal 5
Proof Suppose 119891 isin 119873119899120575
(119892) then
infin
sum
119896=119899+1
1198961003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le 120575
infin
sum
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le
120575
119899 + 1
(43)
Since 119892 isin 119860 we have
infin
sum
119896=119899+1
119887119896le
119887 |119896|
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(44)
Consider10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt
suminfin
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816
1 minus suminfin
119896=119899+1119887119896
le
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1minus 120574 |119887|
= 1 minus 120590
(45)
Therefore 119891 isin 119860120590 for 120590 given by (42)
Theorem 16 If 119892 isin 119861 and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(1 + 119899120572) 119861119899+1
119862119898
119899+1minus 120574 |119887|
(46)
then 119873119899120575
(119892) sub 119861120590
Corollary 17 If 119892 isin 1198601and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (119899 + 1)119898
minus 120574 |119887|)minus1
(120574 |119887| gt 1)
(47)
then 119873119899120575
(119892) sub 119860120590
1
Corollary 18 If 119892 isin 1198611and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (119899 + 1)119898
(1 + 119899120572) (119899 + 1)119898
minus 120574 |119887|
(48)
then 119873119899120575
(119892) sub 119861120590
1
Corollary 19 If 119892 isin 1198602and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (
120578 + 119899
119899)119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
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
The Scientific World Journal 3
where D119898119891(119911) denote the Salagean derivative of order 119898
given by
D119898
119891 (119911) = 119911 minus
infin
sum
119896=119899+1
119896119898
119886119896119911119896
1198611= 119891 isin A (119899) |
10038161003816100381610038161003816100381610038161003816
1
119887
((1 minus 120572)
D119898119891 (119911)
119911
+ 120572[D119898
119891 (119911)]1015840
minus 1)
10038161003816100381610038161003816100381610038161003816
lt 120574
(19)
Similarly on taking 119902 = 2 119904 = 1 1205721
= 120578 minus 1 (120578 gt minus1)1205722= 1 120573
1= 1 one gets
1198602= 119891 isin A (119899) |
1003816100381610038161003816100381610038161003816
1
119887
(119911[119863119898
119891 (119911)]1015840
times ((1 minus 120572)119863119898
119891 (119911)
+ 120572119911[119863119898
119891 (119911)]1015840
)
minus1
)
1003816100381610038161003816100381610038161003816
lt 120574
(20)
where 119863119898
119891(119911) is the operator introduced and studied byAbubaker and Darus [11] given by
119863119898
119891 (119911) = 119911 minus
infin
sum
119896=119899+1
119862119898
119896(
120578 + 119896 minus 1
119896 minus 1) 119911119896
1198612=119891 isin A (119899) |
10038161003816100381610038161003816100381610038161003816
1
119887
((1 minus 120572)
119863119898
119891 (119911)
119911
+ 120572[119863119898
119891 (119911)]1015840
minus 1)
10038161003816100381610038161003816100381610038161003816
lt 120574
(21)
Further by taking 119898 = 0 in the definition of the classes119860 and 119861 we could arrive at 119878
119899(119902 119904 120572 119887 120574) and 119877
119899(119902 119904 120572 119887 120574)
whichwere introduced and studied byMurugusundaramoor-thy et al [12]
In this paper we establish the coefficient inequalitiesfor the classes 119860 and 119861 and several inclusion relationshipsinvolving 119899-120575 neighborhoods of analytic univalent functionswith negative and missing coefficients belonging to theseclasses
2 Coefficient Inequalities
Theorem 3 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860 if and only ifinfin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
119896119861119896119886119896le 120574 |119887| (22)
Proof Let the functions of form (1) belong to the class 119860Then in view of (15) and (16) we get
1003816100381610038161003816100381610038161003816100381610038161003816
suminfin
119896=119899+1([1 + 120572 (119896 minus 1)] minus 119896) 119862
119898
119896119861119896119886119896119911119896
119911 minus suminfin
119896=119899+1[1 + 120572 (119896 minus 1)] 119862
119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
lt 120574 |119887|
suminfin
119896=119899+1([1 + 120572 (119896 minus 1)] minus 119896) 119862
119898
119896119861119896119886119896119903119896minus1
1 minus suminfin
119896=119899+1[1 + 120572 (119896 minus 1)] 119862
119898
119896119861119896119886119896119903119896minus1
lt 120574 |119887|
(23)
Letting 119903 rarr 1minus through real values we get the required
assertion (22) Conversely suppose 119891(119911) satisfies (22) thenin view of (16) consider
100381610038161003816100381610038161003816
119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840
minus (1 minus 120572)D119898
120582120583(1205721 1205731) 119891 (119911)
minus120572119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840100381610038161003816100381610038161003816
minus 120574 |119887|
100381610038161003816100381610038161003816
(1 minus 120572)D119898
120582120583(1205721 1205731) 119891 (119911)
+ 120572119911[D119898
120582120583(1205721 1205731) 119891 (119911)]
1015840100381610038161003816100381610038161003816
=
1003816100381610038161003816100381610038161003816100381610038161003816
infin
sum
119896=119899+1
([1 + 120572 (119896 minus 1)] minus 119896) 119862119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
minus 120574 |119887|
1003816100381610038161003816100381610038161003816100381610038161003816
119911 minus
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896119861119896119886119896119911119896
1003816100381610038161003816100381610038161003816100381610038161003816
lt
infin
sum
119896=119899+1
([1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896)
times 119862119898
119896119861119896119886119896minus 120574 |119887|
le 0
(24)
Hence the result follows
Similarly we prove the following
Theorem 4 Let the function 119891 isin A(119899) be as defined in (1)Then 119891 isin 119861 if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896119861119896119886119896le 120574 |119887| (25)
Corollary 5 Let the function 119891 isin A(119899) as given in (1) Then119891(119911) isin 119860
1if and only if
infin
sum
119896=119899+1
[1+120572 (119896minus1)] (120574 |119887|minus1)+119896 119896119898
119886119896le120574 |119887| (26)
Corollary 6 Let the function 119891 isin A(119899) be as defined in (1)Then 119891 isin 119861
1if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119896119898
119886119896le 120574 |119887| (27)
Corollary 7 Let the function 119891 isin A(119899) as given in (1) Then119891 isin 119860
2if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] (120574 |119887| minus 1) + 119896119862119898
119896
times (
120578 + 119896 minus 1
119896 minus 1) 119886119896le 120574 |119887|
(28)
4 The Scientific World Journal
Corollary 8 Let the function 119891(119911) isin A(119899) be as defined in(1) Then 119891 isin 119861
2if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896(
120578 + 119896 minus 1
119896 minus 1) 119886119896le 120574 |119887| (29)
3 Inclusion Relationships
Theorem 9 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(120574 |119887| gt 1)
(30)
then 119860 sub 119873119899120575
(119890)
Proof Let 119891 isin 119860 Then in view of (22) we have
119862119898
119899+1119861119899+1
[(1+119899120572) (120574 |119887|minus1) 119899+1]
infin
sum
119896=119899+1
119886119896
le 120574 |119887|
(31)
infin
sum
119896=119899+1
119886119896le
120574 |119887|
119862119898
119899+1119861119899+1
[(1 + 119899120572) (120574 |119887| minus 1) 119899 + 1]
(32)
Consider
119862119898
119899+1119861119899+1
infin
sum
119896=119899+1
119896119886119896
le 120574 |119887| + (1 + 119899120572) (1 minus 120574 |119887|) 119861119899+1
119862119898
119899+1
infin
sum
119896=119899+1
119886119896
le 120574 |119887| +
(1 + 119899120572) (1 minus 120574 |119887|) 120574 |119887|
(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1
le
120574 |119887| (119899 + 1)
(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1
(33)
Hence
infin
sum
119896=119899+1
119896119886119896le
120574 |119887| (119899 + 1)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
= 120575
(34)
Hence the result follows
In similar manner we establish the following result
Theorem 10 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(35)
then 119861 sub 119873119899120575
(119890)
Corollary 11 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898 (36)
then 1198601sub 119873119899120575
(119890)
Corollary 12 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) (119899 + 1)119898 (37)
then 1198611sub 119873119899120575
(119890)
Corollary 13 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119862119898
119899+1(120578+119899
119899)
(38)
then 1198602sub 119873119899120575
(119890)
Corollary 14 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) 119862119898
119899+1(120578+119899
119899)
(39)
then 1198612sub 119873119899120575
(119890)
4 Neighborhoods for 119860120590 and 119861
120590
In this section we determine the neighborhood properties of119860120590 and 119861
120590 Here the classes119860120590 consist of functions119891 isin A(119899)
for which there exists a function 119892(119911) isin 119860 such that
10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt 1 minus 120590 (119911 isin Δ 0 le 120590 lt 1) (40)
In the same way we define 119861120590 consisting of functions 119891(119911) isin
A(119899) for which there exists another function 119892(119911) isin 119861 suchthat
10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt 1 minus 120590 (119911 isin Δ 0 le 120590 lt 1) (41)
Theorem 15 If 119892 isin 119860 and
120590 = 1 minus
120575
119899 + 1
times [(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times 119861119899+1
119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times119861119899+1
119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(42)
then 119873119899120575
(119892) sub 119860120590
The Scientific World Journal 5
Proof Suppose 119891 isin 119873119899120575
(119892) then
infin
sum
119896=119899+1
1198961003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le 120575
infin
sum
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le
120575
119899 + 1
(43)
Since 119892 isin 119860 we have
infin
sum
119896=119899+1
119887119896le
119887 |119896|
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(44)
Consider10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt
suminfin
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816
1 minus suminfin
119896=119899+1119887119896
le
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1minus 120574 |119887|
= 1 minus 120590
(45)
Therefore 119891 isin 119860120590 for 120590 given by (42)
Theorem 16 If 119892 isin 119861 and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(1 + 119899120572) 119861119899+1
119862119898
119899+1minus 120574 |119887|
(46)
then 119873119899120575
(119892) sub 119861120590
Corollary 17 If 119892 isin 1198601and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (119899 + 1)119898
minus 120574 |119887|)minus1
(120574 |119887| gt 1)
(47)
then 119873119899120575
(119892) sub 119860120590
1
Corollary 18 If 119892 isin 1198611and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (119899 + 1)119898
(1 + 119899120572) (119899 + 1)119898
minus 120574 |119887|
(48)
then 119873119899120575
(119892) sub 119861120590
1
Corollary 19 If 119892 isin 1198602and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (
120578 + 119899
119899)119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
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 The Scientific World Journal
Corollary 8 Let the function 119891(119911) isin A(119899) be as defined in(1) Then 119891 isin 119861
2if and only if
infin
sum
119896=119899+1
[1 + 120572 (119896 minus 1)] 119862119898
119896(
120578 + 119896 minus 1
119896 minus 1) 119886119896le 120574 |119887| (29)
3 Inclusion Relationships
Theorem 9 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(120574 |119887| gt 1)
(30)
then 119860 sub 119873119899120575
(119890)
Proof Let 119891 isin 119860 Then in view of (22) we have
119862119898
119899+1119861119899+1
[(1+119899120572) (120574 |119887|minus1) 119899+1]
infin
sum
119896=119899+1
119886119896
le 120574 |119887|
(31)
infin
sum
119896=119899+1
119886119896le
120574 |119887|
119862119898
119899+1119861119899+1
[(1 + 119899120572) (120574 |119887| minus 1) 119899 + 1]
(32)
Consider
119862119898
119899+1119861119899+1
infin
sum
119896=119899+1
119896119886119896
le 120574 |119887| + (1 + 119899120572) (1 minus 120574 |119887|) 119861119899+1
119862119898
119899+1
infin
sum
119896=119899+1
119886119896
le 120574 |119887| +
(1 + 119899120572) (1 minus 120574 |119887|) 120574 |119887|
(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1
le
120574 |119887| (119899 + 1)
(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1
(33)
Hence
infin
sum
119896=119899+1
119896119886119896le
120574 |119887| (119899 + 1)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
= 120575
(34)
Hence the result follows
In similar manner we establish the following result
Theorem 10 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(35)
then 119861 sub 119873119899120575
(119890)
Corollary 11 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898 (36)
then 1198601sub 119873119899120575
(119890)
Corollary 12 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) (119899 + 1)119898 (37)
then 1198611sub 119873119899120575
(119890)
Corollary 13 If
120575 =
120574 |119887| (1 + 119899)
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119862119898
119899+1(120578+119899
119899)
(38)
then 1198602sub 119873119899120575
(119890)
Corollary 14 If
120575 =
120574 |119887| (1 + 119899)
(1 + 119899120572) 119862119898
119899+1(120578+119899
119899)
(39)
then 1198612sub 119873119899120575
(119890)
4 Neighborhoods for 119860120590 and 119861
120590
In this section we determine the neighborhood properties of119860120590 and 119861
120590 Here the classes119860120590 consist of functions119891 isin A(119899)
for which there exists a function 119892(119911) isin 119860 such that
10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt 1 minus 120590 (119911 isin Δ 0 le 120590 lt 1) (40)
In the same way we define 119861120590 consisting of functions 119891(119911) isin
A(119899) for which there exists another function 119892(119911) isin 119861 suchthat
10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt 1 minus 120590 (119911 isin Δ 0 le 120590 lt 1) (41)
Theorem 15 If 119892 isin 119860 and
120590 = 1 minus
120575
119899 + 1
times [(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times 119861119899+1
119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times119861119899+1
119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(42)
then 119873119899120575
(119892) sub 119860120590
The Scientific World Journal 5
Proof Suppose 119891 isin 119873119899120575
(119892) then
infin
sum
119896=119899+1
1198961003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le 120575
infin
sum
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le
120575
119899 + 1
(43)
Since 119892 isin 119860 we have
infin
sum
119896=119899+1
119887119896le
119887 |119896|
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(44)
Consider10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt
suminfin
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816
1 minus suminfin
119896=119899+1119887119896
le
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1minus 120574 |119887|
= 1 minus 120590
(45)
Therefore 119891 isin 119860120590 for 120590 given by (42)
Theorem 16 If 119892 isin 119861 and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(1 + 119899120572) 119861119899+1
119862119898
119899+1minus 120574 |119887|
(46)
then 119873119899120575
(119892) sub 119861120590
Corollary 17 If 119892 isin 1198601and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (119899 + 1)119898
minus 120574 |119887|)minus1
(120574 |119887| gt 1)
(47)
then 119873119899120575
(119892) sub 119860120590
1
Corollary 18 If 119892 isin 1198611and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (119899 + 1)119898
(1 + 119899120572) (119899 + 1)119898
minus 120574 |119887|
(48)
then 119873119899120575
(119892) sub 119861120590
1
Corollary 19 If 119892 isin 1198602and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (
120578 + 119899
119899)119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
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
The Scientific World Journal 5
Proof Suppose 119891 isin 119873119899120575
(119892) then
infin
sum
119896=119899+1
1198961003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le 120575
infin
sum
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816le
120575
119899 + 1
(43)
Since 119892 isin 119860 we have
infin
sum
119896=119899+1
119887119896le
119887 |119896|
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
(44)
Consider10038161003816100381610038161003816100381610038161003816
119891 (119911)
119892 (119911)
minus 1
10038161003816100381610038161003816100381610038161003816
lt
suminfin
119896=119899+1
1003816100381610038161003816119886119896minus 119887119896
1003816100381610038161003816
1 minus suminfin
119896=119899+1119887119896
le
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] 119861119899+1
119862119898
119899+1minus 120574 |119887|
= 1 minus 120590
(45)
Therefore 119891 isin 119860120590 for 120590 given by (42)
Theorem 16 If 119892 isin 119861 and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) 119861119899+1
119862119898
119899+1
(1 + 119899120572) 119861119899+1
119862119898
119899+1minus 120574 |119887|
(46)
then 119873119899120575
(119892) sub 119861120590
Corollary 17 If 119892 isin 1198601and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (119899 + 1)119898
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (119899 + 1)119898
minus 120574 |119887|)minus1
(120574 |119887| gt 1)
(47)
then 119873119899120575
(119892) sub 119860120590
1
Corollary 18 If 119892 isin 1198611and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (119899 + 1)119898
(1 + 119899120572) (119899 + 1)119898
minus 120574 |119887|
(48)
then 119873119899120575
(119892) sub 119861120590
1
Corollary 19 If 119892 isin 1198602and
120590 = 1 minus
120575
119899 + 1
[(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1] (
120578 + 119899
119899)119862119898
119899+1
times ([(1 + 119899120572) (120574 |119887| minus 1) + 119899 + 1]
times (
120578 + 119899
119899)119862119898
119899+1minus 120574 |119887|)
minus1
(120574 |119887| gt 1)
(49)
then 119873119899120575
(119892) sub 119860120590
2
Corollary 20 If 119892 isin 1198612and
120590 = 1 minus
120575
119899 + 1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1
(1 + 119899120572) (120578+119899
119899) 119862119898
119899+1minus 120574 |119887|
(50)
then 119873119899120575
(119892) sub 119861120590
2
Conflict of Interests
The authors declare that they do not have conflict of interestsregarding the publication of this paper
References
[1] M Nasr and M Aouf ldquoStarlike functions of complex orderrdquoJournal of Natural Sciences and Mathematics vol 25 no 1 pp1ndash12 1985
[2] P Wiatrowski ldquoOn the coefficients of some family of holomor-phic functionsrdquo Zeszyty Naukowe vol 2 no 39 pp 75ndash85 1970
[3] St Ruscheweyh ldquoNeighborhoods of univalent functionsrdquo Pro-ceedings of the AmericanMathematical Society pp 521ndash527 1981
[4] H Silverman ldquoNeighborhoods of a class of analytic functionsrdquoFar East Journal of Mathematical Sciences vol 3 no 2 pp 165ndash169 1995
[5] N Marikkannan ldquoA subclass of analytic functions anda gener-alised differential operatorrdquo communicated
[6] G S Salagean ldquoSubclasses of univalent functionsrdquo in ComplexAnalysis vol 1013 of Lecture Notes in Mathematics pp 362ndash372 Berlin Germany 1981 Proceedings of the 5th Romanian-Finnish Seminar Part 1 Bucharest Romania 1981
[7] St Ruscheweyh ldquoNew criteria for univalent functionsrdquoProceed-ing of the American Mathematical Society vol 49 pp 109ndash1151975
[8] S Owa ldquoSome applications of the fractional calculusrdquo inProceedings of the Workshop on Fractional Calculus vol 138of Research Notes in Mathematics pp 164ndash175 University ofStrathclyde Ross Priory UK 1985
[9] B C Carlson and S B Shaffer ldquoStarlike and Prestarlike hyper-geometric functionsrdquo SIAM Journal on Mathematical Analysisvol 15 no 4 pp 737ndash745 2002
[10] J Dziok and H M Srivatsava ldquoClasses of analytic func-tions associated with the generalized hypergeometric functionrdquoApplied Mathematics and Computation vol 103 no 1 pp 1ndash131999
6 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
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 The Scientific World Journal
[11] A A A Abubaker and M Darus ldquoNeighborhoods of certainclasses of analytic functions defined by a generalized differentialoperatorrdquo International Journal of Mathematical Analysis vol 4no 45-48 pp 2373ndash2380 2010
[12] G Murugusundaramoorthy T Rosy and S SivasubramanianldquoOn certain classes of analytic functions of complex orderdefined by Dziok-Srivatsava operatorrdquo Journal of MathematicalInequalities 2007
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
