Research ArticleOn Certain Class of Non-BazileviI Functions of Order 120572 + 119894120573Defined by a Differential Subordination

A G Alamoush and M Darus

School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia (UKM)43600 Bangi Selangor Malaysia

Correspondence should be addressed to M Darus maslinaukmedumy

Received 29 April 2014 Revised 27 June 2014 Accepted 2 July 2014 Published 17 July 2014

Academic Editor Salim Messaoudi

Copyright copy 2014 A G Alamoush and M Darus This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We introduce a new subclass 119873119899(120582 120572 120573 119860 119861) of Non-Bazilevic functions of order 120572 + 119894120573 Some subordination relations and

inequality properties are discussed The results obtained generalize the related work of some authors In addition some othernew results are also obtained

1 Introduction

Let 119860119899denote the class of the functions 119891 of the form

119891 (119911) = 119911 +

infin

sum

119896=119899+1

119886119896119911119896

(119899 isin N = 1 2 3 ) (1)

which are analytic in the open unit disk U = 119911 isin C |119911| lt

1 Let 119891(119911) and 119865(119911) be analytic in U Then we say that thefunction 119891(119911) is subordinate to 119865(119911) in U if there exists ananalytic function 119908(119911) in U such that |119908(119911)| le 1 and 119891(119911) =

119865(119908(119911)) denoted 119891 ≺ 119865 or 119891(119911) ≺ 119865(119911) If 119865(119911) is univalentinU then the subordination is equivalent to 119891(0) = 119865(0) and119891(U) sub 119865(U)

Assume that 0 lt 120572 lt 1 a function 119891(119911) isin 119860119899 is in119873(120572)

if and only if

R1198911015840

(119911) (119911

119891(119911))

1+120572

gt 0 (119911 isin U) (2)

The class 119873(120572) was introduced by Obradovic [1] recentlyThis class of functions was said to be of Non-Bazilevic typeTo this date this class was studied in a direction of findingnecessary conditions over 120572 that embeds this class into theclass of univalent functions or its subclasses which is still anopen problem

Assume that 0 lt 120572 lt 1 120582 isin C minus1 le 119861 le 1 119860 = 119861 and119860 isin R we consider the following subclass of 119860

119899

119873(120582 120572 119860 119861) = 119891 (119911) isin 119860119899 (1 + 120582) (

119911

119891 (119911))

120572

minus 1205821199111198911015840

(119911)

119891 (119911)

times(119911

119891 (119911))

120572

≺1 + 119860119911

1 + 119861119911 (119911 isin U)

(3)

where all the powers are principal values and we apply thisagreement to get the following definition

Definition 1 Let 119873(120582 120572 120583) denote the class of functions in119860119899satisfying the inequality

R(1 + 120582) (119911

119891(119911))

120572

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572

gt 120583 (4)

where 0 lt 120572 lt 1 120582 isin C 0 le 120583 lt 1 and 119911 isin U

The classes 119873(120582 120572 119860 119861) and 119873(120582 120572 120583) were studied byWang et al [2]

In the present paper similarly we define the followingclass of analytic functions

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2014 Article ID 458090 6 pageshttpdxdoiorg1011552014458090

2 International Journal of Differential Equations

Definition 2 Let 119873119899(120582 120572 120573 119860 119861) denote the class of func-

tions in 119860119899satisfying the inequality

(1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

≺1 + 119860119911

1 + 119861119911 (119911 isin U)

(5)

where 120582 isin C 120572 ge 0 120573 isin R minus1 le 119861 le 1 119860 = 119861 and 119860 isin RAll the powers in (5) are principal values

We say that the function 119891(119911) in this class is Non-Bazilevic functions of type 120572 + 119894120573

Definition 3 Let 119891(119911) isin 119873119899(120582 120572 120573 120583) if and only if 119891(119911) isin

119860119899and it satisfies

R(1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

gt 120583

(6)

where 120582 isin C 120572 ge 0 120573 isin R 0 le 120583 lt 1 and 119911 isin U

In particular if 120573 = 0 it reduces to the class119873(120582 120572 119860 119861)

studied in [2]If 120573 = 0 120582 = minus1 119899 = 1 119860 = 1 and 119861 = minus1 then the

class 119873119899(120582 120572 120573 119860 119861) reduces to the class of non-Bazilevic

functions If 120573 = 0 120582 = minus1 119899 = 1 119860 = 1 minus 2120583 and 119861 =

minus1 then the class 119873119899(120582 120572 120573 119860 119861) reduces to the class of

non-Bazilevic functions of order 120583 (0 le 120583 lt 1) Tuneskiand Darus studied the Fekete-Szego problem of the class119873(minus1 120572 0 1 minus 2120583 minus1) [3] Other works related to Bazilevicand non-Bazilevic can be found in ([4ndash9])

In the present paper we will discuss the subordi-nation relations and inequality properties of the class119873119899(120582 120572 120573 119860 119861) The results presented here generalize and

improve some known results and some other new results areobtained

2 Some Lemmas

Lemma 4 (see [10]) Let 119865(119911) = 1 + 119887119899119911119899

+ 119887119899+1

119911119899+1

+ sdot sdot sdot beanalytic in U and ℎ(119911) be analytic and convex in U ℎ(0) = 1If

119865 (119911) +1

1198881199111198651015840

(119911) ≺ ℎ (119911) (7)

where 119888 = 0 and Re 119888 ge 0 then

119865 (119911) ≺119888

119899119911minus119888119899

int

119911

0

119905(119888119899)minus1

ℎ (119905) 119889119905 ≺ ℎ (119911) (8)

and (119888119899)119911minus119888119899

int119911

0

119905(119888119899)minus1

ℎ(119905)119889119905 is the best dominant for thedifferential subordination (7)

Lemma 5 (see [11]) Let minus1 le 1198611le 1198612lt 1198602le 1198601le 1 then

1 + 1198602119911

1 + 1198612119911≺1 + 119860

1119911

1 + 1198611119911 (9)

Lemma 6 (see [12]) Let 119865(119911) be analytic and convex in U119891(119911) isin 119860

119899 119892(119911) isin 119860

119899 If

119891 (119911) ≺ 119865 (119911) 119892 (119911) ≺ 119865 (119911) 0 le 120582 le 1 (10)

then

120582119891 (119911) + (1 minus 120582) 119892 (119911) ≺ 119865 (119911) (11)

Lemma 7 (see [13]) Let 119891(119911) = suminfin

119896=1119886119896119911119896 be analytic in U

and119892(119911) = suminfin

119896=1119887119896119911119896 analytic and convex inU If119891(119911) ≺ 119892(119911)

then |119886119896| le |119887119896| for 119896 = 1 2

Lemma 8 Let 120582 isin C 120572 ge 0 120573 isin R 120572 + 119894120573 = 0 minus1 le 119861 le 1119860 = 119861 and 119860 isin R Then 119891(119911) isin 119873

119899(120582 120572 120573 119860 119861) if and only

if

119865 (119911) +120582

120572 + 1198941205731199111198651015840

(119911) ≺1 + 119860119911

1 + 119861119911 (12)

where

119865 (119911) = (119911

119891 (119911))

120572+119894120573

(13)

Proof Let

(119911

119891(119911))

120572+119894120573

= 119865 (119911) (14)

Then by taking the derivatives of both sides of (14) andthrough simple calculation we have

(1 + 120582) (119911

119891 (119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891 (119911))

120572+119894120573

= 119865 (119911) +120582

120572 + 1198941205731199111198651015840

(119911)

(15)

since 119891(119911) isin 119873119899(120582 120572 120573 119860 119861) we have

119865 (119911) +120582

120572 + 1198941205731199111198651015840

(119911) ≺1 + 119860119911

1 + 119861119911 (16)

3 Main Results

Theorem 9 Let 120582 isin C 120572 ge 0 120573 isin R 120572 + 119894120573 = 0 minus1 le 119861 le 1119860 = 119861 and 119860 isin R If 119891(119911) isin 119873

119899(120582 120572 120573 119860 119861) then

(119911

119891(119911))

120572+119894120573

≺120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906 ≺1 + 119860119911

1 + 119861119911

(17)

Proof First let 119865(119911) = (119911119891(119911))120572+119894120573 then 119865(119911) = 1 + 119887

119899119911119899

+

119887119899+1

119911119899+1

+ sdot sdot sdot is analytic in U Now suppose that 119891(119911) isin

119873119899(120582 120572 120573 119860 119861) by Lemma 8 we know that

119865 (119911) +120582

120572 + 1198941205731199111198651015840

(119911) ≺1 + 119860119911

1 + 119861119911 (18)

International Journal of Differential Equations 3

It is obvious that ℎ(119911) = (1 + 119860119911)(1 + 119861119911) is analytic andconvex in U ℎ(0) = 1 Since 120572 + 119894120573 = 0 120572 ge 0 120582 = 0 andR(120572 + 119894120573)120582 ge 0 therefore it follows from Lemma 4 that

(119911

119891 (119911))

120572+119894120573

= 119865 (119911)

≺120572 + 119894120573

120582119899119911minus(120572+119894120573)120582119899

int

119911

0

119905((120572+119894120573)120582119899)minus1

ℎ (119905) 119889119905

=120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906 ≺1 + 119860119911

1 + 119861119911

(19)

Corollary 10 Let 120582 isin C 120572 ge 0 120573 isin R 120572+119894120573 = 0 and 120583 = 1If 119891(119911) isin 119860

119899satisfies

(1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

≺1 + (1 minus 2120583) 119911

1 minus 119911(119911 isin U)

(20)

then

(119911

119891(119911))

120572+119894120573

≺120572 + 119894120573

120582119899

times int

1

0

1 + (1 minus 2120583) 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119905

(119911 isin U)

(21)

or equivalent to

(119911

119891(119911))

120572+119894120573

≺ 120583 +(1 minus 120583) (120572 + 119894120573)

120582119899

times int

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119905 (119911 isin U)

(22)

Corollary 11 Let 120582 isin C 120572 ge 0 120573 isin R 120572 + 119894120573 = 0 andR120582 ge 0 then

119873119899(120582 120572 120573 119860 119861) sub 119873

119899(0 120572 120573 119860 119861) (23)

Theorem 12 Let 0 le 1205821le 1205822 120572 ge 0 120573 isin R 120572 + 119894120573 = 0 and

minus1 le 1198611le 1198612lt 1198602le 1198601le 1 then

119873119899(1205822 120572 120573 119860

2 1198612) sub 119873

119899(1205821 120572 120573 119860

1 1198611) (24)

Proof Suppose that 119891(119911) isin 119873119899(1205822 120572 120573 119860

2 1198612) we have

119891(119911) isin 119860119899 and

(1 + 1205822) (

119911

119891(119911))

120572+119894120573

minus 1205822

1199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

≺1 + 119860

2119911

1 + 1198612119911

(119911 isin U)

(25)

Since minus1 le 1198611le 1198612lt 1198602le 1198601le 1 therefore it follows from

Lemma 5 that

(1 + 1205822) (

119911

119891(119911))

120572+119894120573

minus 1205822

1199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

≺1 + 119860

1119911

1 + 1198611119911

(119911 isin U)

(26)

that is 119891(119911) isin 119873119899(1205822 120572 120573 119860

1 1198611) So Theorem 12 is proved

when 1205821= 1205822ge 0

When 1205822gt 1205821ge 0 then we can see from Corollary 11

that 119891(119911) isin 119873119899(0 120572 120573 119860

1 1198611) then

(119911

119891(119911))

120572+119894120573

≺1 + 119860

1119911

1 + 1198611119911 (27)

But

(1 + 1205821) (

119911

119891 (119911))

120572+119894120573

minus 1205821

1199111198911015840

(119911)

119891 (119911)(

119911

119891 (119911))

120572+119894120573

= (1 minus1205821

1205822

)(119911

119891 (119911))

120572+119894120573

+1205821

1205822

times [(1 + 1205822) (

119911

119891 (119911))

120572+119894120573

minus 1205822

1199111198911015840

(119911)

119891 (119911)(

119911

119891 (119911))

120572+119894120573

]

(28)

It is obvious that ℎ1(119911) = (1 + 119860

1119911)(1 + 119861

1119911) is analytic

and convex inU So we obtain from Lemma 6 and differentialsubordinations (26) and (27) that

(1 + 1205821) (

119911

119891 (119911))

120572+119894120573

minus 1205821

1199111198911015840

(119911)

119891 (119911)(

119911

119891 (119911))

120572+119894120573

≺1 + 119860

1119911

1 + 1198611119911

(29)

that is 119891(119911) isin 119873119899(1205821 120572 120573 119860

1 1198611) Thus we have

119873119899(1205822 120572 120573 119860

2 1198612) sub 119873

119899(1205821 120572 120573 119860

1 1198611) (30)

Corollary 13 Let 0 le 1205821le 1205822 0 le 120583

1le 1205832lt 1 120572 ge 0 120573 isin

R and 120572 + 119894120573 = 0 then

119873119899(1205822 120572 120573 120583

2) sub 119873

119899(1205821 120572 120573 120583

1) (31)

Theorem 14 Let 120582 isin C 120572 ge 0 120573 isin R 120572 + 119894120573 = 0 minus1 le

119861 le 1 119860 = 119861 and 119860 isin R If 119891(119911) isin 119873119899(120582 120572 120573 119860 119861) then

inf119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572+119894120573

lt sup119911isinU

R120572 + 119894120573

120582119899

timesint

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(32)

4 International Journal of Differential Equations

Proof Suppose that 119891(119911) isin 119873n(120582 120572 120573 119860 119861) then fromTheorem 9 we know that

(119911

119891(119911))

120572+119894120573

≺120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906 (33)

Therefore from the definition of the subordination we have

R(119911

119891(119911))

120572+119894120573

gt inf119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

R(119911

119891(119911))

120572+119894120573

lt sup119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(34)

Corollary 15 Let 120582 isin C 120572 ge 0 120573 isin R 120572+119894120573 = 0 and 120583 lt

1 If 119891(119911) isin 119873119899(120582 120572 120573 1 minus 2120583 minus1) then

120583 + (1 minus 120583) inf119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572+119894120573

lt 120583 + (1 minus 120583) sup119911isinU

R120572 + 119894120573

120582119899

timesint

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(35)

Corollary 16 Let 120582 isin C 120572 ge 0 120573 isin R 120572+119894120573 = 0 and 120583 gt

1 If 119891(119911) isin 119860119899 then

R((1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

) lt 120583

(119911 isin U)

(36)

then

120583 + (1 minus 120583) sup119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572+119894120573

lt 120583 + (1 minus 120583) inf119911isinU

R120572 + 119894120573

120582119899

timesint

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(37)

Corollary 17 Let 120582 isin C 120572 ge 0 and minus 1 le 119861 lt 119860 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(38)

and inequality (38) is sharp with the extremal function definedby

119891120582120572119861119860

(119911) = 119911(120572

120582119899int

1

0

1 + 119860119911119899

119906

1 + 119861119911119899119906119906(120572120582119899)minus1

119889119906)

minus(1120572)

(39)

Proof Suppose that119891(119911) isin 119873119899(120582 120572 0 119860 119861) fromTheorem 9

we know

(119911

119891(119911))

120572

≺120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906 (40)

Therefore from the definition of the subordination and 119860 gt

119861 we have that

R(119911

119891(119911))

120572

lt sup119911isinU

R120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906

le120572

120582119899int

1

0

sup119911isinU

1 + 119860119906119911

1 + 119861119906119911 119906(120572120582119899)minus1

119889119906

lt120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906

R(119911

119891(119911))

120572

gt inf119911isinU

R120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906

ge120572

120582119899int

1

0

inf119911isinU

1 + 119860119906119911

1 + 119861119906119911 u(120572120582119899)minus1119889119906

gt120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906

(41)

It is obvious that inequality (38) is sharp with the extremalfunction given by (39)

Corollary 18 Let 120582 isin C 120572 ge 0 and 120583 lt 1 If 119891(119911) isin

119873119899(120582 120572 0 1 minus 2120583 minus1) then

120572

120582119899int

1

0

1 minus (1 minus 2120583) 119906

1 + 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 + (1 minus 2120583) 119906

1 minus 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(42)

International Journal of Differential Equations 5

and inequality (42) is equivalent to

120583 +(1 minus 120583) 120572

120582119899int

1

0

1 minus 119906

1 + 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt 120583 +(1 minus 120583) 120572

120582119899int

1

0

1 + 119906

1 minus 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(43)

The inequality (42) is sharp with the extremal function definedby

119891120582120572120573

(119911) = 119911(120572

120582119899int

1

0

1 + (1 minus 2120573)119911119899

119906

1 minus 119911119899119906119906(120572120582119899)minus1

119889119906)

minus1120572

119889119911

(44)

Corollary 19 Let 120582 isin C 120572 ge 0 and minus1 le 119860 lt 119861 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(45)

and inequality (45) is sharp with the extremal function givenby (39)

Proof Applying similar method as in Corollary 17 we get theresult

Corollary 20 Let 120582 isin C 120572 ge 0 and 120583 gt 1 If 119891(119911) isin 119860119899

satisfies

R((1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

) lt 120583

(119911 isin U)

(46)

then

120572

120582119899int

1

0

1 + (1 minus 2120583) 119906

1 minus 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 minus (1 minus 2120583) 119906

1 + 119906119906(120572120582119899)minus1

119889119906

(47)

and inequality (47) is equivalent to

120583 +(1 minus 120583) 120572

120582119899int

1

0

1 + 119906

1 minus 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt 120583 +(1 minus 120583) 120572

120582119899int

1

0

1 minus 119906

1 + 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(48)

and inequality (47) is sharp with the extremal function definedby equality (44)

IfR119908 ge 0 then (R119908)12

le R11990812

le R11991112 (see [2 12])

So we have the following

Corollary 21 Let 120582 isin C 120572 ge 0 and minus1 le 119861 lt 119860 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

(120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906)

12

lt R[(119911

119891(119911))

120572

]

12

lt (120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906)

12

(119911 isin U)

(49)

and inequality (49) is sharp with the extremal function definedby equality (39)

Proof FromTheorem 9 we have

(119911

119891(119911))

120572

≺1 + 119860119911

1 + 119861119911 (50)

Since minus1 le 119860 lt 119861 le 1 we have

0 le1 minus 119860

1 minus 119861lt R(

119911

119891(119911))

120572

lt1 + 119860

1 + 119861 (51)

Thus from inequality (38) we can get inequality (49) Itis obvious that inequality (49) is sharp with the extremalfunction defined by equality (39)

Corollary 22 Let 120582 isin C 120572 ge 0 and minus 1 le 119860 lt 119861 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

(120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906)

12

lt R[(119911

119891(119911))

120572

]

12

lt (120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906)

12

(119911 isin U)

(52)

and inequality (52) is sharp with the extremal function definedby equality (39)

6 International Journal of Differential Equations

Proof Applying similar method as in Corollary 21 we get therequired result

Remark 23 From Corollaries 21 and 22 we can generalizethe corresponding results and some other special classes ofanalytic functions

Corollary 24 Let 120582 isin C 120572 ge 0 minus1 le 119861 lt 119860 le 1 and 119860 isin Rif 119891(119911) = 119911 + sum

infin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572 0 119860 119861) then one has

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572|(53)

and inequality (53) is sharp with the extremal function definedby equality (39)

Proof Suppose that 119891(119911) = 119911 + suminfin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572

0 119860 119861) then we have

(1 + 120582) (119911

119891(119911))

120572

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572

= 1 + (minus119899120582 minus 120572) 119886119899+1

119911119899

+ sdot sdot sdot ≺1 + 119860119911

1 + 119861119911

(54)

It follows from Lemma 7 that

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572| (55)

Thus we can get (53) Notice that

119891 (119911) = 119911 +119860 minus 119861

119899120582 + 120572119911119899+1

+ sdot sdot sdot isin 119873119899(120582 120572 0 119860 119861) (56)

we obtain that the inequality (53) is sharp

Remark 25 Setting 120582 = 119899 = 119860 = 1 and 119861 = minus1 inCorollary 24 we get the results obtained by [14]

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia under the Grant AP-2013-009 The authors alsowould like to thank the referees for the comments andsuggestions to improve the paper

References

[1] M Obradovic ldquoA class of univalent functionsrdquo HokkaidoMathematical Journal vol 27 no 2 pp 329ndash335 1998

[2] Z Wang C Gao and M Liao ldquoOn certain generalized class ofnon-Bazilevic functionsrdquo Acta Mathematica Academiae Paeda-gogicae Nyıregyhaziensis vol 21 no 2 pp 147ndash154 2005

[3] N Tuneski and M Darus ldquoFekete-Szego functional for non-Bazilevic functionsrdquo Acta Mathematica Academiae Paedagogi-cae Nyı regyhaziensis vol 18 pp 63ndash65 2002

[4] A A Amer andM Darus ldquoDistortion theorem for certain classof Bazilevic functionsrdquo International Journal of MathematicalAnalysis vol 6 no 9ndash12 pp 591ndash597 2012

[5] R W Ibrahim and M Darus ldquoMixed class of functions ofnon-Bazilevic type and bounded turningrdquo Far East Journal ofMathematical Sciences vol 67 no 1 pp 141ndash152 2012

[6] R W Ibrahim and M Darus ldquoArgument estimate for non-Bazilevic type and bounded turning functionsrdquo Far East Journalof Mathematical Sciences vol 68 no 2 pp 175ndash183 2012

[7] R W Ibrahim M Darus and N Tuneski ldquoOn subordinationfor classes of non-Bazilevic typerdquo Annales Universitatis MariaeCurie-Sklodowska Lublin-Polonia A vol 64 no 2 pp 49ndash602010

[8] M Darus and R W Ibrahim ldquoOn subclasses of uniformlyBazilevic type functions involving generalised differential andintegral operatorsrdquo Far East Journal of Mathematical Sciences(FJMS) vol 33 no 3 pp 401ndash411 2009

[9] T N Shanmugam S Sivasubramanian M Darus and SKavitha ldquoOn sandwich theorems for certain subclasses ofnon-Bazilevic functions involving Cho-Kim transformationrdquoComplex Variables and Elliptic Equations vol 52 no 10-11 pp1017ndash1028 2007

[10] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981

[11] M S Liu ldquoOn a subclass of 119901-valent close-to-convex functionsof order 120573 and type 120572rdquo Journal of Mathematical Study vol 30no 1 pp 102ndash104 1997

[12] M-S Liu ldquoOn certain class of analytic functions defined bydifferential subordinationrdquo Acta Mathematica Scientia B vol22 no 3 pp 388ndash392 2002

[13] W Rogosinski ldquoOn the coefficients of subordination functionsrdquoProceedings of the London Mathematical Society 2 vol 48 pp48ndash82 1943

[14] R Singh ldquo On Bazilevic functionsrdquo Proceedings of the AmericanMathematical Society vol 38 pp 261ndash271 1973

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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OptimizationJournal of

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Differential Equations 3

It is obvious that ℎ(119911) = (1 + 119860119911)(1 + 119861119911) is analytic andconvex in U ℎ(0) = 1 Since 120572 + 119894120573 = 0 120572 ge 0 120582 = 0 andR(120572 + 119894120573)120582 ge 0 therefore it follows from Lemma 4 that

(119911

119891 (119911))

120572+119894120573

= 119865 (119911)

≺120572 + 119894120573

120582119899119911minus(120572+119894120573)120582119899

int

119911

0

119905((120572+119894120573)120582119899)minus1

ℎ (119905) 119889119905

=120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906 ≺1 + 119860119911

1 + 119861119911

(19)

Corollary 10 Let 120582 isin C 120572 ge 0 120573 isin R 120572+119894120573 = 0 and 120583 = 1If 119891(119911) isin 119860

119899satisfies

(1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

≺1 + (1 minus 2120583) 119911

1 minus 119911(119911 isin U)

(20)

then

(119911

119891(119911))

120572+119894120573

≺120572 + 119894120573

120582119899

times int

1

0

1 + (1 minus 2120583) 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119905

(119911 isin U)

(21)

or equivalent to

(119911

119891(119911))

120572+119894120573

≺ 120583 +(1 minus 120583) (120572 + 119894120573)

120582119899

times int

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119905 (119911 isin U)

(22)

Corollary 11 Let 120582 isin C 120572 ge 0 120573 isin R 120572 + 119894120573 = 0 andR120582 ge 0 then

119873119899(120582 120572 120573 119860 119861) sub 119873

119899(0 120572 120573 119860 119861) (23)

Theorem 12 Let 0 le 1205821le 1205822 120572 ge 0 120573 isin R 120572 + 119894120573 = 0 and

minus1 le 1198611le 1198612lt 1198602le 1198601le 1 then

119873119899(1205822 120572 120573 119860

2 1198612) sub 119873

119899(1205821 120572 120573 119860

1 1198611) (24)

Proof Suppose that 119891(119911) isin 119873119899(1205822 120572 120573 119860

2 1198612) we have

119891(119911) isin 119860119899 and

(1 + 1205822) (

119911

119891(119911))

120572+119894120573

minus 1205822

1199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

≺1 + 119860

2119911

1 + 1198612119911

(119911 isin U)

(25)

Since minus1 le 1198611le 1198612lt 1198602le 1198601le 1 therefore it follows from

Lemma 5 that

(1 + 1205822) (

119911

119891(119911))

120572+119894120573

minus 1205822

1199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

≺1 + 119860

1119911

1 + 1198611119911

(119911 isin U)

(26)

that is 119891(119911) isin 119873119899(1205822 120572 120573 119860

1 1198611) So Theorem 12 is proved

when 1205821= 1205822ge 0

When 1205822gt 1205821ge 0 then we can see from Corollary 11

that 119891(119911) isin 119873119899(0 120572 120573 119860

1 1198611) then

(119911

119891(119911))

120572+119894120573

≺1 + 119860

1119911

1 + 1198611119911 (27)

But

(1 + 1205821) (

119911

119891 (119911))

120572+119894120573

minus 1205821

1199111198911015840

(119911)

119891 (119911)(

119911

119891 (119911))

120572+119894120573

= (1 minus1205821

1205822

)(119911

119891 (119911))

120572+119894120573

+1205821

1205822

times [(1 + 1205822) (

119911

119891 (119911))

120572+119894120573

minus 1205822

1199111198911015840

(119911)

119891 (119911)(

119911

119891 (119911))

120572+119894120573

]

(28)

It is obvious that ℎ1(119911) = (1 + 119860

1119911)(1 + 119861

1119911) is analytic

and convex inU So we obtain from Lemma 6 and differentialsubordinations (26) and (27) that

(1 + 1205821) (

119911

119891 (119911))

120572+119894120573

minus 1205821

1199111198911015840

(119911)

119891 (119911)(

119911

119891 (119911))

120572+119894120573

≺1 + 119860

1119911

1 + 1198611119911

(29)

that is 119891(119911) isin 119873119899(1205821 120572 120573 119860

1 1198611) Thus we have

119873119899(1205822 120572 120573 119860

2 1198612) sub 119873

119899(1205821 120572 120573 119860

1 1198611) (30)

Corollary 13 Let 0 le 1205821le 1205822 0 le 120583

1le 1205832lt 1 120572 ge 0 120573 isin

R and 120572 + 119894120573 = 0 then

119873119899(1205822 120572 120573 120583

2) sub 119873

119899(1205821 120572 120573 120583

1) (31)

Theorem 14 Let 120582 isin C 120572 ge 0 120573 isin R 120572 + 119894120573 = 0 minus1 le

119861 le 1 119860 = 119861 and 119860 isin R If 119891(119911) isin 119873119899(120582 120572 120573 119860 119861) then

inf119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572+119894120573

lt sup119911isinU

R120572 + 119894120573

120582119899

timesint

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(32)

4 International Journal of Differential Equations

Proof Suppose that 119891(119911) isin 119873n(120582 120572 120573 119860 119861) then fromTheorem 9 we know that

(119911

119891(119911))

120572+119894120573

≺120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906 (33)

Therefore from the definition of the subordination we have

R(119911

119891(119911))

120572+119894120573

gt inf119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

R(119911

119891(119911))

120572+119894120573

lt sup119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(34)

Corollary 15 Let 120582 isin C 120572 ge 0 120573 isin R 120572+119894120573 = 0 and 120583 lt

1 If 119891(119911) isin 119873119899(120582 120572 120573 1 minus 2120583 minus1) then

120583 + (1 minus 120583) inf119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572+119894120573

lt 120583 + (1 minus 120583) sup119911isinU

R120572 + 119894120573

120582119899

timesint

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(35)

Corollary 16 Let 120582 isin C 120572 ge 0 120573 isin R 120572+119894120573 = 0 and 120583 gt

1 If 119891(119911) isin 119860119899 then

R((1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

) lt 120583

(119911 isin U)

(36)

then

120583 + (1 minus 120583) sup119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572+119894120573

lt 120583 + (1 minus 120583) inf119911isinU

R120572 + 119894120573

120582119899

timesint

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(37)

Corollary 17 Let 120582 isin C 120572 ge 0 and minus 1 le 119861 lt 119860 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(38)

and inequality (38) is sharp with the extremal function definedby

119891120582120572119861119860

(119911) = 119911(120572

120582119899int

1

0

1 + 119860119911119899

119906

1 + 119861119911119899119906119906(120572120582119899)minus1

119889119906)

minus(1120572)

(39)

Proof Suppose that119891(119911) isin 119873119899(120582 120572 0 119860 119861) fromTheorem 9

we know

(119911

119891(119911))

120572

≺120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906 (40)

Therefore from the definition of the subordination and 119860 gt

119861 we have that

R(119911

119891(119911))

120572

lt sup119911isinU

R120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906

le120572

120582119899int

1

0

sup119911isinU

1 + 119860119906119911

1 + 119861119906119911 119906(120572120582119899)minus1

119889119906

lt120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906

R(119911

119891(119911))

120572

gt inf119911isinU

R120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906

ge120572

120582119899int

1

0

inf119911isinU

1 + 119860119906119911

1 + 119861119906119911 u(120572120582119899)minus1119889119906

gt120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906

(41)

It is obvious that inequality (38) is sharp with the extremalfunction given by (39)

Corollary 18 Let 120582 isin C 120572 ge 0 and 120583 lt 1 If 119891(119911) isin

119873119899(120582 120572 0 1 minus 2120583 minus1) then

120572

120582119899int

1

0

1 minus (1 minus 2120583) 119906

1 + 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 + (1 minus 2120583) 119906

1 minus 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(42)

International Journal of Differential Equations 5

and inequality (42) is equivalent to

120583 +(1 minus 120583) 120572

120582119899int

1

0

1 minus 119906

1 + 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt 120583 +(1 minus 120583) 120572

120582119899int

1

0

1 + 119906

1 minus 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(43)

The inequality (42) is sharp with the extremal function definedby

119891120582120572120573

(119911) = 119911(120572

120582119899int

1

0

1 + (1 minus 2120573)119911119899

119906

1 minus 119911119899119906119906(120572120582119899)minus1

119889119906)

minus1120572

119889119911

(44)

Corollary 19 Let 120582 isin C 120572 ge 0 and minus1 le 119860 lt 119861 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(45)

and inequality (45) is sharp with the extremal function givenby (39)

Proof Applying similar method as in Corollary 17 we get theresult

Corollary 20 Let 120582 isin C 120572 ge 0 and 120583 gt 1 If 119891(119911) isin 119860119899

satisfies

R((1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

) lt 120583

(119911 isin U)

(46)

then

120572

120582119899int

1

0

1 + (1 minus 2120583) 119906

1 minus 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 minus (1 minus 2120583) 119906

1 + 119906119906(120572120582119899)minus1

119889119906

(47)

and inequality (47) is equivalent to

120583 +(1 minus 120583) 120572

120582119899int

1

0

1 + 119906

1 minus 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt 120583 +(1 minus 120583) 120572

120582119899int

1

0

1 minus 119906

1 + 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(48)

and inequality (47) is sharp with the extremal function definedby equality (44)

IfR119908 ge 0 then (R119908)12

le R11990812

le R11991112 (see [2 12])

So we have the following

Corollary 21 Let 120582 isin C 120572 ge 0 and minus1 le 119861 lt 119860 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

(120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906)

12

lt R[(119911

119891(119911))

120572

]

12

lt (120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906)

12

(119911 isin U)

(49)

and inequality (49) is sharp with the extremal function definedby equality (39)

Proof FromTheorem 9 we have

(119911

119891(119911))

120572

≺1 + 119860119911

1 + 119861119911 (50)

Since minus1 le 119860 lt 119861 le 1 we have

0 le1 minus 119860

1 minus 119861lt R(

119911

119891(119911))

120572

lt1 + 119860

1 + 119861 (51)

Thus from inequality (38) we can get inequality (49) Itis obvious that inequality (49) is sharp with the extremalfunction defined by equality (39)

Corollary 22 Let 120582 isin C 120572 ge 0 and minus 1 le 119860 lt 119861 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

(120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906)

12

lt R[(119911

119891(119911))

120572

]

12

lt (120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906)

12

(119911 isin U)

(52)

and inequality (52) is sharp with the extremal function definedby equality (39)

6 International Journal of Differential Equations

Proof Applying similar method as in Corollary 21 we get therequired result

Remark 23 From Corollaries 21 and 22 we can generalizethe corresponding results and some other special classes ofanalytic functions

Corollary 24 Let 120582 isin C 120572 ge 0 minus1 le 119861 lt 119860 le 1 and 119860 isin Rif 119891(119911) = 119911 + sum

infin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572 0 119860 119861) then one has

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572|(53)

and inequality (53) is sharp with the extremal function definedby equality (39)

Proof Suppose that 119891(119911) = 119911 + suminfin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572

0 119860 119861) then we have

(1 + 120582) (119911

119891(119911))

120572

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572

= 1 + (minus119899120582 minus 120572) 119886119899+1

119911119899

+ sdot sdot sdot ≺1 + 119860119911

1 + 119861119911

(54)

It follows from Lemma 7 that

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572| (55)

Thus we can get (53) Notice that

119891 (119911) = 119911 +119860 minus 119861

119899120582 + 120572119911119899+1

+ sdot sdot sdot isin 119873119899(120582 120572 0 119860 119861) (56)

we obtain that the inequality (53) is sharp

Remark 25 Setting 120582 = 119899 = 119860 = 1 and 119861 = minus1 inCorollary 24 we get the results obtained by [14]

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia under the Grant AP-2013-009 The authors alsowould like to thank the referees for the comments andsuggestions to improve the paper

References

[1] M Obradovic ldquoA class of univalent functionsrdquo HokkaidoMathematical Journal vol 27 no 2 pp 329ndash335 1998

[2] Z Wang C Gao and M Liao ldquoOn certain generalized class ofnon-Bazilevic functionsrdquo Acta Mathematica Academiae Paeda-gogicae Nyıregyhaziensis vol 21 no 2 pp 147ndash154 2005

[3] N Tuneski and M Darus ldquoFekete-Szego functional for non-Bazilevic functionsrdquo Acta Mathematica Academiae Paedagogi-cae Nyı regyhaziensis vol 18 pp 63ndash65 2002

[4] A A Amer andM Darus ldquoDistortion theorem for certain classof Bazilevic functionsrdquo International Journal of MathematicalAnalysis vol 6 no 9ndash12 pp 591ndash597 2012

[5] R W Ibrahim and M Darus ldquoMixed class of functions ofnon-Bazilevic type and bounded turningrdquo Far East Journal ofMathematical Sciences vol 67 no 1 pp 141ndash152 2012

[6] R W Ibrahim and M Darus ldquoArgument estimate for non-Bazilevic type and bounded turning functionsrdquo Far East Journalof Mathematical Sciences vol 68 no 2 pp 175ndash183 2012

[7] R W Ibrahim M Darus and N Tuneski ldquoOn subordinationfor classes of non-Bazilevic typerdquo Annales Universitatis MariaeCurie-Sklodowska Lublin-Polonia A vol 64 no 2 pp 49ndash602010

[8] M Darus and R W Ibrahim ldquoOn subclasses of uniformlyBazilevic type functions involving generalised differential andintegral operatorsrdquo Far East Journal of Mathematical Sciences(FJMS) vol 33 no 3 pp 401ndash411 2009

[9] T N Shanmugam S Sivasubramanian M Darus and SKavitha ldquoOn sandwich theorems for certain subclasses ofnon-Bazilevic functions involving Cho-Kim transformationrdquoComplex Variables and Elliptic Equations vol 52 no 10-11 pp1017ndash1028 2007

[10] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981

[11] M S Liu ldquoOn a subclass of 119901-valent close-to-convex functionsof order 120573 and type 120572rdquo Journal of Mathematical Study vol 30no 1 pp 102ndash104 1997

[12] M-S Liu ldquoOn certain class of analytic functions defined bydifferential subordinationrdquo Acta Mathematica Scientia B vol22 no 3 pp 388ndash392 2002

[13] W Rogosinski ldquoOn the coefficients of subordination functionsrdquoProceedings of the London Mathematical Society 2 vol 48 pp48ndash82 1943

[14] R Singh ldquo On Bazilevic functionsrdquo Proceedings of the AmericanMathematical Society vol 38 pp 261ndash271 1973

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 International Journal of Differential Equations

Proof Suppose that 119891(119911) isin 119873n(120582 120572 120573 119860 119861) then fromTheorem 9 we know that

(119911

119891(119911))

120572+119894120573

≺120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906 (33)

Therefore from the definition of the subordination we have

R(119911

119891(119911))

120572+119894120573

gt inf119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

R(119911

119891(119911))

120572+119894120573

lt sup119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119860119911119906

1 + 119861119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(34)

Corollary 15 Let 120582 isin C 120572 ge 0 120573 isin R 120572+119894120573 = 0 and 120583 lt

1 If 119891(119911) isin 119873119899(120582 120572 120573 1 minus 2120583 minus1) then

120583 + (1 minus 120583) inf119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572+119894120573

lt 120583 + (1 minus 120583) sup119911isinU

R120572 + 119894120573

120582119899

timesint

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(35)

Corollary 16 Let 120582 isin C 120572 ge 0 120573 isin R 120572+119894120573 = 0 and 120583 gt

1 If 119891(119911) isin 119860119899 then

R((1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

) lt 120583

(119911 isin U)

(36)

then

120583 + (1 minus 120583) sup119911isinU

R120572 + 119894120573

120582119899int

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572+119894120573

lt 120583 + (1 minus 120583) inf119911isinU

R120572 + 119894120573

120582119899

timesint

1

0

1 + 119911119906

1 minus 119911119906119906((120572+119894120573)120582119899)minus1

119889119906

(37)

Corollary 17 Let 120582 isin C 120572 ge 0 and minus 1 le 119861 lt 119860 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(38)

and inequality (38) is sharp with the extremal function definedby

119891120582120572119861119860

(119911) = 119911(120572

120582119899int

1

0

1 + 119860119911119899

119906

1 + 119861119911119899119906119906(120572120582119899)minus1

119889119906)

minus(1120572)

(39)

Proof Suppose that119891(119911) isin 119873119899(120582 120572 0 119860 119861) fromTheorem 9

we know

(119911

119891(119911))

120572

≺120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906 (40)

Therefore from the definition of the subordination and 119860 gt

119861 we have that

R(119911

119891(119911))

120572

lt sup119911isinU

R120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906

le120572

120582119899int

1

0

sup119911isinU

1 + 119860119906119911

1 + 119861119906119911 119906(120572120582119899)minus1

119889119906

lt120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906

R(119911

119891(119911))

120572

gt inf119911isinU

R120572

120582119899int

1

0

1 + 119860119906119911

1 + 119861119906119911119906(120572120582119899)minus1

119889119906

ge120572

120582119899int

1

0

inf119911isinU

1 + 119860119906119911

1 + 119861119906119911 u(120572120582119899)minus1119889119906

gt120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906

(41)

It is obvious that inequality (38) is sharp with the extremalfunction given by (39)

Corollary 18 Let 120582 isin C 120572 ge 0 and 120583 lt 1 If 119891(119911) isin

119873119899(120582 120572 0 1 minus 2120583 minus1) then

120572

120582119899int

1

0

1 minus (1 minus 2120583) 119906

1 + 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 + (1 minus 2120583) 119906

1 minus 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(42)

International Journal of Differential Equations 5

and inequality (42) is equivalent to

120583 +(1 minus 120583) 120572

120582119899int

1

0

1 minus 119906

1 + 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt 120583 +(1 minus 120583) 120572

120582119899int

1

0

1 + 119906

1 minus 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(43)

The inequality (42) is sharp with the extremal function definedby

119891120582120572120573

(119911) = 119911(120572

120582119899int

1

0

1 + (1 minus 2120573)119911119899

119906

1 minus 119911119899119906119906(120572120582119899)minus1

119889119906)

minus1120572

119889119911

(44)

Corollary 19 Let 120582 isin C 120572 ge 0 and minus1 le 119860 lt 119861 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(45)

and inequality (45) is sharp with the extremal function givenby (39)

Proof Applying similar method as in Corollary 17 we get theresult

Corollary 20 Let 120582 isin C 120572 ge 0 and 120583 gt 1 If 119891(119911) isin 119860119899

satisfies

R((1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

) lt 120583

(119911 isin U)

(46)

then

120572

120582119899int

1

0

1 + (1 minus 2120583) 119906

1 minus 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 minus (1 minus 2120583) 119906

1 + 119906119906(120572120582119899)minus1

119889119906

(47)

and inequality (47) is equivalent to

120583 +(1 minus 120583) 120572

120582119899int

1

0

1 + 119906

1 minus 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt 120583 +(1 minus 120583) 120572

120582119899int

1

0

1 minus 119906

1 + 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(48)

and inequality (47) is sharp with the extremal function definedby equality (44)

IfR119908 ge 0 then (R119908)12

le R11990812

le R11991112 (see [2 12])

So we have the following

Corollary 21 Let 120582 isin C 120572 ge 0 and minus1 le 119861 lt 119860 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

(120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906)

12

lt R[(119911

119891(119911))

120572

]

12

lt (120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906)

12

(119911 isin U)

(49)

and inequality (49) is sharp with the extremal function definedby equality (39)

Proof FromTheorem 9 we have

(119911

119891(119911))

120572

≺1 + 119860119911

1 + 119861119911 (50)

Since minus1 le 119860 lt 119861 le 1 we have

0 le1 minus 119860

1 minus 119861lt R(

119911

119891(119911))

120572

lt1 + 119860

1 + 119861 (51)

Thus from inequality (38) we can get inequality (49) Itis obvious that inequality (49) is sharp with the extremalfunction defined by equality (39)

Corollary 22 Let 120582 isin C 120572 ge 0 and minus 1 le 119860 lt 119861 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

(120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906)

12

lt R[(119911

119891(119911))

120572

]

12

lt (120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906)

12

(119911 isin U)

(52)

and inequality (52) is sharp with the extremal function definedby equality (39)

6 International Journal of Differential Equations

Proof Applying similar method as in Corollary 21 we get therequired result

Remark 23 From Corollaries 21 and 22 we can generalizethe corresponding results and some other special classes ofanalytic functions

Corollary 24 Let 120582 isin C 120572 ge 0 minus1 le 119861 lt 119860 le 1 and 119860 isin Rif 119891(119911) = 119911 + sum

infin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572 0 119860 119861) then one has

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572|(53)

and inequality (53) is sharp with the extremal function definedby equality (39)

Proof Suppose that 119891(119911) = 119911 + suminfin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572

0 119860 119861) then we have

(1 + 120582) (119911

119891(119911))

120572

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572

= 1 + (minus119899120582 minus 120572) 119886119899+1

119911119899

+ sdot sdot sdot ≺1 + 119860119911

1 + 119861119911

(54)

It follows from Lemma 7 that

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572| (55)

Thus we can get (53) Notice that

119891 (119911) = 119911 +119860 minus 119861

119899120582 + 120572119911119899+1

+ sdot sdot sdot isin 119873119899(120582 120572 0 119860 119861) (56)

we obtain that the inequality (53) is sharp

Remark 25 Setting 120582 = 119899 = 119860 = 1 and 119861 = minus1 inCorollary 24 we get the results obtained by [14]

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia under the Grant AP-2013-009 The authors alsowould like to thank the referees for the comments andsuggestions to improve the paper

References

[1] M Obradovic ldquoA class of univalent functionsrdquo HokkaidoMathematical Journal vol 27 no 2 pp 329ndash335 1998

[2] Z Wang C Gao and M Liao ldquoOn certain generalized class ofnon-Bazilevic functionsrdquo Acta Mathematica Academiae Paeda-gogicae Nyıregyhaziensis vol 21 no 2 pp 147ndash154 2005

[3] N Tuneski and M Darus ldquoFekete-Szego functional for non-Bazilevic functionsrdquo Acta Mathematica Academiae Paedagogi-cae Nyı regyhaziensis vol 18 pp 63ndash65 2002

[4] A A Amer andM Darus ldquoDistortion theorem for certain classof Bazilevic functionsrdquo International Journal of MathematicalAnalysis vol 6 no 9ndash12 pp 591ndash597 2012

[5] R W Ibrahim and M Darus ldquoMixed class of functions ofnon-Bazilevic type and bounded turningrdquo Far East Journal ofMathematical Sciences vol 67 no 1 pp 141ndash152 2012

[6] R W Ibrahim and M Darus ldquoArgument estimate for non-Bazilevic type and bounded turning functionsrdquo Far East Journalof Mathematical Sciences vol 68 no 2 pp 175ndash183 2012

[7] R W Ibrahim M Darus and N Tuneski ldquoOn subordinationfor classes of non-Bazilevic typerdquo Annales Universitatis MariaeCurie-Sklodowska Lublin-Polonia A vol 64 no 2 pp 49ndash602010

[8] M Darus and R W Ibrahim ldquoOn subclasses of uniformlyBazilevic type functions involving generalised differential andintegral operatorsrdquo Far East Journal of Mathematical Sciences(FJMS) vol 33 no 3 pp 401ndash411 2009

[9] T N Shanmugam S Sivasubramanian M Darus and SKavitha ldquoOn sandwich theorems for certain subclasses ofnon-Bazilevic functions involving Cho-Kim transformationrdquoComplex Variables and Elliptic Equations vol 52 no 10-11 pp1017ndash1028 2007

[10] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981

[11] M S Liu ldquoOn a subclass of 119901-valent close-to-convex functionsof order 120573 and type 120572rdquo Journal of Mathematical Study vol 30no 1 pp 102ndash104 1997

[12] M-S Liu ldquoOn certain class of analytic functions defined bydifferential subordinationrdquo Acta Mathematica Scientia B vol22 no 3 pp 388ndash392 2002

[13] W Rogosinski ldquoOn the coefficients of subordination functionsrdquoProceedings of the London Mathematical Society 2 vol 48 pp48ndash82 1943

[14] R Singh ldquo On Bazilevic functionsrdquo Proceedings of the AmericanMathematical Society vol 38 pp 261ndash271 1973

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

International Journal of Differential Equations 5

and inequality (42) is equivalent to

120583 +(1 minus 120583) 120572

120582119899int

1

0

1 minus 119906

1 + 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt 120583 +(1 minus 120583) 120572

120582119899int

1

0

1 + 119906

1 minus 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(43)

The inequality (42) is sharp with the extremal function definedby

119891120582120572120573

(119911) = 119911(120572

120582119899int

1

0

1 + (1 minus 2120573)119911119899

119906

1 minus 119911119899119906119906(120572120582119899)minus1

119889119906)

minus1120572

119889119911

(44)

Corollary 19 Let 120582 isin C 120572 ge 0 and minus1 le 119860 lt 119861 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(45)

and inequality (45) is sharp with the extremal function givenby (39)

Proof Applying similar method as in Corollary 17 we get theresult

Corollary 20 Let 120582 isin C 120572 ge 0 and 120583 gt 1 If 119891(119911) isin 119860119899

satisfies

R((1 + 120582) (119911

119891(119911))

120572+119894120573

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572+119894120573

) lt 120583

(119911 isin U)

(46)

then

120572

120582119899int

1

0

1 + (1 minus 2120583) 119906

1 minus 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt120572

120582119899int

1

0

1 minus (1 minus 2120583) 119906

1 + 119906119906(120572120582119899)minus1

119889119906

(47)

and inequality (47) is equivalent to

120583 +(1 minus 120583) 120572

120582119899int

1

0

1 + 119906

1 minus 119906119906(120572120582119899)minus1

119889119906

lt R(119911

119891(119911))

120572

lt 120583 +(1 minus 120583) 120572

120582119899int

1

0

1 minus 119906

1 + 119906119906(120572120582119899)minus1

119889119906 (119911 isin U)

(48)

and inequality (47) is sharp with the extremal function definedby equality (44)

IfR119908 ge 0 then (R119908)12

le R11990812

le R11991112 (see [2 12])

So we have the following

Corollary 21 Let 120582 isin C 120572 ge 0 and minus1 le 119861 lt 119860 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

(120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906)

12

lt R[(119911

119891(119911))

120572

]

12

lt (120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906)

12

(119911 isin U)

(49)

and inequality (49) is sharp with the extremal function definedby equality (39)

Proof FromTheorem 9 we have

(119911

119891(119911))

120572

≺1 + 119860119911

1 + 119861119911 (50)

Since minus1 le 119860 lt 119861 le 1 we have

0 le1 minus 119860

1 minus 119861lt R(

119911

119891(119911))

120572

lt1 + 119860

1 + 119861 (51)

Thus from inequality (38) we can get inequality (49) Itis obvious that inequality (49) is sharp with the extremalfunction defined by equality (39)

Corollary 22 Let 120582 isin C 120572 ge 0 and minus 1 le 119860 lt 119861 le 1 If119891(119911) isin 119873

119899(120582 120572 0 119860 119861) then

(120572

120582119899int

1

0

1 + 119860119906

1 + 119861119906119906(120572120582119899)minus1

119889119906)

12

lt R[(119911

119891(119911))

120572

]

12

lt (120572

120582119899int

1

0

1 minus 119860119906

1 minus 119861119906119906(120572120582119899)minus1

119889119906)

12

(119911 isin U)

(52)

and inequality (52) is sharp with the extremal function definedby equality (39)

6 International Journal of Differential Equations

Proof Applying similar method as in Corollary 21 we get therequired result

Remark 23 From Corollaries 21 and 22 we can generalizethe corresponding results and some other special classes ofanalytic functions

Corollary 24 Let 120582 isin C 120572 ge 0 minus1 le 119861 lt 119860 le 1 and 119860 isin Rif 119891(119911) = 119911 + sum

infin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572 0 119860 119861) then one has

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572|(53)

and inequality (53) is sharp with the extremal function definedby equality (39)

Proof Suppose that 119891(119911) = 119911 + suminfin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572

0 119860 119861) then we have

(1 + 120582) (119911

119891(119911))

120572

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572

= 1 + (minus119899120582 minus 120572) 119886119899+1

119911119899

+ sdot sdot sdot ≺1 + 119860119911

1 + 119861119911

(54)

It follows from Lemma 7 that

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572| (55)

Thus we can get (53) Notice that

119891 (119911) = 119911 +119860 minus 119861

119899120582 + 120572119911119899+1

+ sdot sdot sdot isin 119873119899(120582 120572 0 119860 119861) (56)

we obtain that the inequality (53) is sharp

Remark 25 Setting 120582 = 119899 = 119860 = 1 and 119861 = minus1 inCorollary 24 we get the results obtained by [14]

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia under the Grant AP-2013-009 The authors alsowould like to thank the referees for the comments andsuggestions to improve the paper

References

[1] M Obradovic ldquoA class of univalent functionsrdquo HokkaidoMathematical Journal vol 27 no 2 pp 329ndash335 1998

[2] Z Wang C Gao and M Liao ldquoOn certain generalized class ofnon-Bazilevic functionsrdquo Acta Mathematica Academiae Paeda-gogicae Nyıregyhaziensis vol 21 no 2 pp 147ndash154 2005

[3] N Tuneski and M Darus ldquoFekete-Szego functional for non-Bazilevic functionsrdquo Acta Mathematica Academiae Paedagogi-cae Nyı regyhaziensis vol 18 pp 63ndash65 2002

[4] A A Amer andM Darus ldquoDistortion theorem for certain classof Bazilevic functionsrdquo International Journal of MathematicalAnalysis vol 6 no 9ndash12 pp 591ndash597 2012

[5] R W Ibrahim and M Darus ldquoMixed class of functions ofnon-Bazilevic type and bounded turningrdquo Far East Journal ofMathematical Sciences vol 67 no 1 pp 141ndash152 2012

[6] R W Ibrahim and M Darus ldquoArgument estimate for non-Bazilevic type and bounded turning functionsrdquo Far East Journalof Mathematical Sciences vol 68 no 2 pp 175ndash183 2012

[7] R W Ibrahim M Darus and N Tuneski ldquoOn subordinationfor classes of non-Bazilevic typerdquo Annales Universitatis MariaeCurie-Sklodowska Lublin-Polonia A vol 64 no 2 pp 49ndash602010

[8] M Darus and R W Ibrahim ldquoOn subclasses of uniformlyBazilevic type functions involving generalised differential andintegral operatorsrdquo Far East Journal of Mathematical Sciences(FJMS) vol 33 no 3 pp 401ndash411 2009

[9] T N Shanmugam S Sivasubramanian M Darus and SKavitha ldquoOn sandwich theorems for certain subclasses ofnon-Bazilevic functions involving Cho-Kim transformationrdquoComplex Variables and Elliptic Equations vol 52 no 10-11 pp1017ndash1028 2007

[10] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981

[11] M S Liu ldquoOn a subclass of 119901-valent close-to-convex functionsof order 120573 and type 120572rdquo Journal of Mathematical Study vol 30no 1 pp 102ndash104 1997

[12] M-S Liu ldquoOn certain class of analytic functions defined bydifferential subordinationrdquo Acta Mathematica Scientia B vol22 no 3 pp 388ndash392 2002

[13] W Rogosinski ldquoOn the coefficients of subordination functionsrdquoProceedings of the London Mathematical Society 2 vol 48 pp48ndash82 1943

[14] R Singh ldquo On Bazilevic functionsrdquo Proceedings of the AmericanMathematical Society vol 38 pp 261ndash271 1973

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 International Journal of Differential Equations

Proof Applying similar method as in Corollary 21 we get therequired result

Remark 23 From Corollaries 21 and 22 we can generalizethe corresponding results and some other special classes ofanalytic functions

Corollary 24 Let 120582 isin C 120572 ge 0 minus1 le 119861 lt 119860 le 1 and 119860 isin Rif 119891(119911) = 119911 + sum

infin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572 0 119860 119861) then one has

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572|(53)

and inequality (53) is sharp with the extremal function definedby equality (39)

Proof Suppose that 119891(119911) = 119911 + suminfin

119896=119899+1119886119896119911119896

isin 119873119899(120582 120572

0 119860 119861) then we have

(1 + 120582) (119911

119891(119911))

120572

minus 1205821199111198911015840

(119911)

119891 (119911)(

119911

119891(119911))

120572

= 1 + (minus119899120582 minus 120572) 119886119899+1

119911119899

+ sdot sdot sdot ≺1 + 119860119911

1 + 119861119911

(54)

It follows from Lemma 7 that

1003816100381610038161003816119886119899+11003816100381610038161003816 le

|119860 minus 119861|

|119899120582 + 120572| (55)

Thus we can get (53) Notice that

119891 (119911) = 119911 +119860 minus 119861

119899120582 + 120572119911119899+1

+ sdot sdot sdot isin 119873119899(120582 120572 0 119860 119861) (56)

we obtain that the inequality (53) is sharp

Remark 25 Setting 120582 = 119899 = 119860 = 1 and 119861 = minus1 inCorollary 24 we get the results obtained by [14]

Conflict of Interests

The authors declare that they have no conflict of interests

Authorsrsquo Contribution

Both authors read and approved the final paper

Acknowledgments

The authors would like to acknowledge and appreciatethe financial support received from Universiti KebangsaanMalaysia under the Grant AP-2013-009 The authors alsowould like to thank the referees for the comments andsuggestions to improve the paper

References

[1] M Obradovic ldquoA class of univalent functionsrdquo HokkaidoMathematical Journal vol 27 no 2 pp 329ndash335 1998

[2] Z Wang C Gao and M Liao ldquoOn certain generalized class ofnon-Bazilevic functionsrdquo Acta Mathematica Academiae Paeda-gogicae Nyıregyhaziensis vol 21 no 2 pp 147ndash154 2005

[3] N Tuneski and M Darus ldquoFekete-Szego functional for non-Bazilevic functionsrdquo Acta Mathematica Academiae Paedagogi-cae Nyı regyhaziensis vol 18 pp 63ndash65 2002

[4] A A Amer andM Darus ldquoDistortion theorem for certain classof Bazilevic functionsrdquo International Journal of MathematicalAnalysis vol 6 no 9ndash12 pp 591ndash597 2012

[5] R W Ibrahim and M Darus ldquoMixed class of functions ofnon-Bazilevic type and bounded turningrdquo Far East Journal ofMathematical Sciences vol 67 no 1 pp 141ndash152 2012

[6] R W Ibrahim and M Darus ldquoArgument estimate for non-Bazilevic type and bounded turning functionsrdquo Far East Journalof Mathematical Sciences vol 68 no 2 pp 175ndash183 2012

[7] R W Ibrahim M Darus and N Tuneski ldquoOn subordinationfor classes of non-Bazilevic typerdquo Annales Universitatis MariaeCurie-Sklodowska Lublin-Polonia A vol 64 no 2 pp 49ndash602010

[8] M Darus and R W Ibrahim ldquoOn subclasses of uniformlyBazilevic type functions involving generalised differential andintegral operatorsrdquo Far East Journal of Mathematical Sciences(FJMS) vol 33 no 3 pp 401ndash411 2009

[9] T N Shanmugam S Sivasubramanian M Darus and SKavitha ldquoOn sandwich theorems for certain subclasses ofnon-Bazilevic functions involving Cho-Kim transformationrdquoComplex Variables and Elliptic Equations vol 52 no 10-11 pp1017ndash1028 2007

[10] S S Miller and P T Mocanu ldquoDifferential subordinations andunivalent functionsrdquo The Michigan Mathematical Journal vol28 no 2 pp 157ndash172 1981

[11] M S Liu ldquoOn a subclass of 119901-valent close-to-convex functionsof order 120573 and type 120572rdquo Journal of Mathematical Study vol 30no 1 pp 102ndash104 1997

[12] M-S Liu ldquoOn certain class of analytic functions defined bydifferential subordinationrdquo Acta Mathematica Scientia B vol22 no 3 pp 388ndash392 2002

[13] W Rogosinski ldquoOn the coefficients of subordination functionsrdquoProceedings of the London Mathematical Society 2 vol 48 pp48ndash82 1943

[14] R Singh ldquo On Bazilevic functionsrdquo Proceedings of the AmericanMathematical Society vol 38 pp 261ndash271 1973

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