Research Article A New Subclass of Analytic Functions Defined by Generalized Ruscheweyh Differential Operator

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1 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 28, Article ID , 12 pages doi:1.1155/28/ Research Article A New Subclass of Analytic Functions Defined by Generalized Ruscheweyh Differential Operator Serap Bulut Civil Aviation College, Kocaeli University, Arslanbey Campus, İzmit-Kocaeli, Turkey Correspondence should be addressed to Serap Bulut, serap.bulut@kocaeli.edu.tr Received 1 July 28; Accepted 3 September 28 Recommended by Narendra Kumar Govil We investigate a new subclass of analytic functions in the open unit disk U which is defined by generalized Ruscheweyh differential operator. Coefficient inequalities, extreme points, and the integral means inequalities for the fractional derivatives of order p η p n, η<1 of functions belonging to this subclass are obtained. Copyright q 28 Serap Bulut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Throughout this paper, we use the following notations: N : {1, 2, 3,...}, N : N {}, R 1 : {u R : u> 1}, R 1 : R 1 \{}. 1.1 Let A denote the class of all functions of the form fz z a n z n, 1.2 which are analytic in the open unit disk U : {z C : z < 1}. For f j Agiven by f j z z a n,j z n j 1, 2, 1.3

2 2 Journal of Inequalities and Applications the Hadamard product or convolution f 1 f 2 of f 1 and f 2 is defined by f 1 f 2 z z a n,1 a n,2 z n. 1.4 Using the convolution 1.4, Shaqsi and Darus 1 introduced the generalization of the Ruscheweyh derivative as follows. For f A,λ, and u R 1, we consider R u z λfz 1 z u1 R λfz z U, 1.5 where R λ fz 1 λfzλzf z, z U. If f Ais of the form 1.2, then we obtain the power series expansion of the form R u λ fz z 1 n 1λCu, na n z n, 1.6 where Cu, n 1 u n 1 n 1! n N, 1.7 and where a n is the Pochhammer symbol or shifted factorial defined in terms of the Gamma function by a n : Γa n Γa { 1, if n, a C \{}, aa 1 a n 1, if n N, a C. 1.8 In the case m N, we have R m λ fz fz m zzm 1, 1.9 m! and for λ, we obtain uth Ruscheweyh derivative introduced in 2, R m Rm. Using the generalized Ruscheweyh derivative operator R u, we define the following λ classes. Definition 1.1. Let S λ u, v; α be the class of functions f Asatisfying { R u λ Re fz } R v λ fz >α 1.1 for some α<1, u R 1, v R 1, λ, and all z U.

3 Serap Bulut 3 In this paper, basic properties of the class S λ u, v; α are studied, such as coefficient bounds, extreme points, and integral means inequalities for the fractional derivative. 2. Coefficient inequalities Theorem 2.1. Let α<1, u R 1,v R 1, and λ. Iff Asatisfies B n u, v, α a n 21 α, 2.1 where B n u, v, α : 1 n 1λ{ Cu, n 1 αcv, n Cu, n1 αcv, n}, 2.2 then f S λ u, v; α. Proof. Let 2.1 be true for α<1, u R 1,v R 1, andλ. For f A, define the function F by Fz : Ru λ fz R v α. fz 2.3 λ It is sufficient to show that Fz 1 Fz1 < for z U. So, we have Fz 1 Fz1 R u λ fz 1 αrv λ fz R u λ fz1 αrv λ fz α 1 n 1λCu, n 1 αcv, na nz n 1 2 α 1 n 1λCu, n1 αcv, na nz n 1 < α 1 n 1λ Cu, n 1 αcv, n a n z n 1 2 α 1 n 1λCu, n1 αcv, n a n z n 1 α 1 n 1λ Cu, n 1 αcv, n a n 2 α 1 n 1λCu, n1 αcv, n a n 2.5 < 1 by 2.1. Therefore, f S λ u, v; α.

4 4 Journal of Inequalities and Applications Theorem 2.2. If f S λ u, v; α,then n 1 a n 1 n u 1λCv, n u a n u 1 n 1λ Cu, n Cv, n u1 2.6 for n 2, witha 1 1. Proof. Define the function Gz : ( 1 R u λ fz ) 1 α R v λ fz α : 1 â n z n. n1 2.7 Since Re{Gz} >, we get â n for n 1, 2,... From the definition of Gz,weobtain R u λ fz αrv λ fz [ R v fz 1 â n z ]. n α n1 So, by 1.6, we have z 1 λ 1 α Cu, 2 αcv, 2a 2z 2 1 2λ 1 α Cu, 3 αcv, 3a 3z 3 z â 1 z 2 â 2 z 3 â 3 z 4 1 λcv, 2a 2 z 2 1 λcv, 2a 2 â 1 z 3 1 λcv, 2a 2 â 2 z λCv, 3a 3 z 3 1 2λCv, 3a 3 â 1 z 4 or z 1 λ 1 α Cu, 2 Cv, 2a 2z 2 1 2λ 1 α Cu, 3 Cv, 3a 3z 3 z â 1 z 2 1λCv, 2a 2 â 1 â 2 z λCv, 3a 3 â 1 1 λcv, 2a 2 â 2 â 3 z 4

5 Serap Bulut 5 or, equivalently, z 1 n 1λ Cu, j Cv, ja n z n 1 α z ( n 1 1 n u 1λCv, n ua n u â u )z n. u When we consider the coefficients of z n of both series in the above equality, we have 1 α n 1 a n 1 n u 1λCv, n ua n u â u. 1 n 1λCu, n Cv, n u Therefore, 1 α n 1 a n 1 n u 1λCv, n u a n u â u 1 n 1λ Cu, n Cv, n u1 n 1 1 n u 1λCv, n u a n u, 1 n 1λ Cu, n Cv, n u since â u 2, u 1, 2, Extreme points Definition 3.1. Let S λ u, v; α be the subclass of S λ u, v; α which consists of function fz z a n z n a n 3.1 whose coefficients satisfy inequality 2.1. Theorem 3.2. Let f 1 z z and f k z z B k u, v, α zk k 2, 3,..., 3.2 where B k u, v, α is given by 2.2.

6 6 Journal of Inequalities and Applications Then f S λ u, v; α if and only if it can be expressed in the form fz δ k f k z, k1 3.3 where δ k and k1 δ k 1. Proof. Assume that fz δ k f k z. k1 3.4 Then fz δ 1 f 1 z δ 1 z k1 δ k f k z ( ) δ k z B k u, v, α zk ( δ k )z δ k B k u, v, α zk 3.5 z δ k B k u, v, α zk. Thus δ k B k u, v, α B ku, v, α δ k 1 δ Therefore, we have f S λ u, v; α. Conversely, suppose that f S λ u, v; α. Since a k B k u, v, α k 2, 3,..., 3.7 we can set δ k : B ku, v, α a k δ 1 : 1 δ k. k 2, 3,..., 3.8

7 Serap Bulut 7 Then fz z k1 a k z k ( δ k )z δ 1 z δ 1 f 1 z δ k B k u, v, α zk ( ) δ k z B k u, v, α zk δ k f k z δ k f k z. k1 3.9 This completes the proof of Theorem 3.2. Corollary 3.3. The extreme points of S λ u, v; α are given by f 1 z z, f k z z where B k u, v, α is given by 2.2. B k u, v, α zk k 2, 3,..., The main integral means inequalities for the fractional derivative We discuss the integral means inequalities for functions f S λ u, v; α. The following definitions of fractional derivatives by Owa 3 also by Srivastava and Owa 4 will be required in our investigation. Definition 4.1. The fractional derivative of order η is defined, for a function f, by Dzfz η 1 d Γ1 η dz z fξ η dξ z ξ η<1, 4.1 where the function f is analytic in a simply connected region of the complex z-plane containing the origin, and the multiplicity of z ξ η is removed by requiring logz ξ to be real when z ξ>. Definition 4.2. Under the hypothesis of Definition 4.1, the fractional derivative of order p η is defined, for a function f,by D pη z fz dp dz p Dη zfz, 4.2 where η<1andp N.

8 8 Journal of Inequalities and Applications It readily follows from 4.1 in Definition 4.1 that D η zz k Γk 1 Γk 1 η zk η η<1, k N. 4.3 We will also need the concept of subordination between analytic functions and a subordination theorem of Littlewood 5 in our investigation. Definition 4.3. Given two functions f and g, which are analytic in U, the function f is said to be subordinate to g in U if there exists a function w analytic in U with w, wz < 1 z U, 4.4 such that fz gwz z U. 4.5 We denote this subordination by fz gz. 4.6 Lemma 4.4. If the functions f and g are analytic in U with fz gz, 4.7 then, for μ> and z re iθ <r<1, fz μ dθ gz μ dθ. 4.8 Our main theorem is contained in the following. Theorem 4.5. Let f S λ u, v; α and suppose that Γk 1Γ3 η p n p p1 a n B k u, v, αγk 1 η pγ2 p 4.9 for p n, k p, η<1, wheren p p1 denotes the Pochhammer symbol defined by n p p1 n pn p 1 n. 4.1 Also let the function f k be defined by f k z z B k u, v, α zk k 2, 3,

9 Serap Bulut 9 If there exists an analytic function w defined by wz k 1 : B ku, v, αγk 1 η p Γk 1 n p p1 Ψna n z n with Ψn Γn p, Γn 1 η p η<1, n 2, 3,..., 4.13 then, for μ> and z re iθ <r<1, pη D fz μ dθ z pη D z f k z μ dθ, η< Proof. By means of 4.3 and Definition 4.2,wefindfrom3.1 that [ D pη z 1 η p z fz 1 Γ2 η p [ z 1 η p 1 Γ2 η p ] Γ2 η pγn 1 a n z n 1 Γn 1 η p Γ2 η pn p p1 Ψna n z ], n where Ψn Γn p, Γn 1 η p η<1, n 2, 3, Since Ψ is a decreasing function of n, weget < Ψn Ψ2 Γ2 p Γ3 η p Similarly, from 4.11, 4.3,andDefinition 4.2, we have D pη z f k z z 1 η p [ 1 Γ2 η p B k u, v, α Γ2 η pγk 1 z ]. k Γk 1 η p

10 1 Journal of Inequalities and Applications For μ>andz re iθ <r<1, we want to show that 1 Γ2 η pn p p1 Ψna n z μdθ n 1 1 B k u, v, α Γ2 η pγk 1 z μdθ. k 1 Γk 1 η p 4.19 So, by applying Lemma 4.4, it is enough to show that 1 Γ2 η pn p p1 Ψna n z n 1 1 B k u, v, α Γ2 η pγk 1 z k 1. Γk 1 η p 4.2 If the above subordination holds true, then we have an analytic function w with w and wz < 1 such that 1 Γ2 η pn p p1 Ψna n z n 1 1 B k u, v, α Γ2 η pγk 1 wz k 1. Γk 1 η p 4.21 By the condition of the theorem, we define the function w by wz k 1 B ku, v, αγk 1 η p Γk 1 n p p1 Ψna n z n 1, 4.22 which readily yields w. For such a function w, we have wz k 1 B ku, v, αγk 1 η p Γk 1 n p p1 Ψna n z n 1 z B ku, v, αγk 1 η p Γk 1 z B ku, v, αγk 1 η p Γk 1 Ψ2 n p p1 a n Γ2 p n p Γ3 η p p1 a n 4.23 z < 1 by means of the hypothesis of the theorem. Thus the theorem is proved. As a special case p, we have the following result from Theorem 4.5.

11 Serap Bulut 11 Corollary 4.6. Let f S λ u, v; α and suppose that na n Γk 1Γ3 η B k u, v, αγk 1 η k 2, 3, If there exists an analytic function w defined by wz k 1 B ku, v, αγk 1 η Γk 1 nψna n z n with Ψn Γn, η<1, n 2, 3,..., 4.26 Γn 1 η then, for μ> and z re iθ <r<1, η D zfz μ dθ η D zf k z μ dθ, η< Letting p 1inTheorem 4.5, we have the following. Corollary 4.7. Let f S λ u, v; α and suppose that nn 1a n Γk 1Γ2 η B k u, v, αγk η k 2, 3, If there exists an analytic function w defined by wz k 1 B ku, v, αγk η Γk 1 nn 1Ψna n z n with Ψn Γn 1, η<1, n 2, 3,..., 4.3 Γn η then, for μ> and z re iθ <r<1, D 1η z fz μ dθ D 1η z f k z μ dθ, η<

12 12 Journal of Inequalities and Applications References 1 K. A. Shaqsi and M. Darus, On univalent functions with respect to K-symmetric points given by a generalised Ruscheweyh derivatives operator, submitted. 2 S. Ruscheweyh, New criteria for univalent functions, Proceedings of the American Mathematical Society, vol. 49, no. 1, pp , S. Owa, On the distortion theorems. I, Kyungpook Mathematical Journal, vol. 18, no. 1, pp , H. M. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series in Mathematics and Its Applications, Ellis Horwood, Chichester, UK; John Wiley & Sons, New York, NY, USA, J. E. Littlewood, On inequalities in the theory of functions, Proceedings of the London Mathematical Society, vol. 23, no. 1, pp , 1925.