Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 134932, 12 pages doi:10.1155/2008/134932 Research Article A New Subclass of Analytic Functions Defined by Generalized Ruscheweyh Differential Operator Serap Bulut ˙ Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Izmit-Kocaeli, Turkey Correspondence should be addressed to Serap Bulut, serap.bulut@kocaeli.edu.tr Received July 2008; Accepted September 2008 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 < of functions belonging to this subclass are obtained Copyright q 2008 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 Introduction Throughout this paper, we use the following notations: N : {1, 2, 3, }, N0 : N ∪ {0}, R−1 : {u ∈ R : u > −1}, 1.1 R0 : R−1 \ {0} −1 Let A denote the class of all functions of the form ∞ f z z an zn , 1.2 n which are analytic in the open unit disk U : {z ∈ C : |z| < 1} For fj ∈ A given by ∞ fj z z n an,j zn j 1, , 1.3 Journal of Inequalities and Applications the Hadamard product or convolution f1 ∗f2 of f1 and f2 is defined by ∞ f1 ∗f2 z z an,1 an,2 zn 1.4 n Using the convolution 1.4 , Shaqsi and Darus introduced the generalization of the Ruscheweyh derivative as follows For f ∈ A, λ ≥ 0, and u ∈ R−1 , we consider z Ru f z λ 1−z u ∗Rλ f z z∈U , 1.5 where Rλ f z − λ f z λzf z , z ∈ U If f ∈ A is of the form 1.2 , then we obtain the power series expansion of the form ∞ Ru f z λ z n − λ C u, n an zn , 1.6 n where u n−1 n−1 ! C u, n 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, a a ··· a if n 0, a ∈ C \ {0}, if n ∈ N, a ∈ C n−1 , 1.8 , 1.9 In the case m ∈ N0 , we have Rm f z λ z zm−1 f z m! m and for λ 0, we obtain uth Ruscheweyh derivative introduced in , Rm Rm Using the generalized Ruscheweyh derivative operator Ru , we define the following λ classes Definition 1.1 Let Sλ u, v; α be the class of functions f ∈ A satisfying Re Ru f z λ Rv f z λ >α for some ≤ α < 1, u ∈ R0 , v ∈ R−1 , λ ≥ 0, and all z ∈ U −1 1.10 Serap Bulut In this paper, basic properties of the class Sλ u, v; α are studied, such as coefficient bounds, extreme points, and integral means inequalities for the fractional derivative Coefficient inequalities Theorem 2.1 Let ≤ α < 1, u ∈ R0 , v ∈ R−1 , and λ ≥ If f ∈ A satisfies −1 ∞ Bn u, v, α |an | ≤ − α , 2.1 n where Bn u, v, α : n − λ {|C u, n − α C v, n | C u, n − α C v, n }, 2.2 then f ∈ Sλ u, v; α Proof Let 2.1 be true for ≤ α < 1, u ∈ R0 , v ∈ R−1 , and λ ≥ For f ∈ A, define the −1 function F by F z : Ru f z λ − α Rv f z λ 2.3 It is sufficient to show that F z −1 and z 4.7 reiθ < r < , 2π 2π |f z |μ dθ ≤ |g z |μ dθ 4.8 Our main theorem is contained in the following Theorem 4.5 Let f ∈ Sλ u, v; α and suppose that ∞ n−p p an n ≤ 1−α Γ k Γ 3−η−p Bk u, v, α Γ k − η − p Γ − p for ≤ p ≤ n, k ≥ p, ≤ η < 1, where n − p n−p p p 4.9 denotes the Pochhammer symbol defined by n−p n−p · · · n 4.10 Also let the function fk be defined by fk z z 1−α zk Bk u, v, α k 2, 3, 4.11 Serap Bulut If there exists an analytic function w defined by k−1 w z Bk u, v, α Γ k − η − p 1−α Γ k : ∞ n−p p 1Ψ n an zn−1 4.12 n with Ψ n Γn Γ n−p , 1−η−p ≤ η < 1, n 2, 3, , 4.13 0≤η and z 2π p η Dz f z μ dθ ≤ 2π p η Dz fk z μ dθ, Proof By means of 4.3 and Definition 4.2, we find from 3.1 that p η Dz f z z1−η−p Γ 2−η−p ∞ n Γ 2−η−p Γ n an zn−1 Γ n 1−η−p 4.15 ∞ 1−η−p z Γ 2−η−p Γ 2−η−p n−p p 1Ψ n an zn−1 , n where Ψ n Γ n−p , Γ n 1−η−p ≤ η < 1, n 2, 3, 4.16 Since Ψ is a decreasing function of n, we get 0 and z ∞ 2π μ Γ 2−η−p n−p n−1 dθ Ψ n an z p n ≤ 4.19 μ 2π − α Γ − η − p Γ k k−1 z dθ Bk u, v, α Γ k 1−η−p So, by applying Lemma 4.4, it is enough to show that ∞ Γ 2−η−p n−p p 1Ψ n an zn−1 ≺ n 2 − α Γ − η − p Γ k k−1 z Bk u, v, α Γ k 1−η−p 4.20 If the above subordination holds true, then we have an analytic function w with w |w z | < such that ∞ Γ 2−η−p n−p p 1Ψ n an zn−1 n 2 1−α Γ 2−η−p Γ k w z Bk u, v, α Γ k 1−η−p and k−1 4.21 By the condition of the theorem, we define the function w by w z Bk u, v, α Γ k − η − p 1−α Γ k k−1 which readily yields w |w z |k−1 ≤ |z| n−p p 1Ψ n an zn−1 , ∞ n−p p 1Ψ n an |z|n−1 n Bk u, v, α Γ k − η − p Ψ 2 1−α Γ k ∞ n−p p an 4.23 n Bk u, v, α Γ k − η − p Γ − p 1−α Γ k Γ 3−η−p ∞ n−p p an n ≤ |z| < by means of the hypothesis of the theorem Thus the theorem is proved As a special case p 4.22 n For such a function w, we have Bk u, v, α Γ k − η − p 1−α Γ k ≤ |z| ∞ 0, we have the following result from Theorem 4.5 Serap Bulut 11 Corollary 4.6 Let f ∈ Sλ u, v; α and suppose that ∞ nan ≤ n 2 1−α Γ k Γ 3−η Bk u, v, α Γ k − η k 2, 3, 4.24 If there exists an analytic function w defined by Bk u, v, α Γ k − η 1−α Γ k k−1 w z ∞ nΨ n an zn−1 4.25 n with Γn , Γ n 1−η Ψ n 2, 3, , 4.26 0≤η and z 2π η Dz f z μ dθ ≤ Letting p ≤ η < 1, n 2π η Dz fk z μ dθ, in Theorem 4.5, we have the following Corollary 4.7 Let f ∈ Sλ u, v; α and suppose that ∞ n n − an ≤ n 2 1−α Γ k Γ 2−η Bk u, v, α Γ k − η 2, 3, If there exists an analytic function w defined by w z k−1 Bk u, v, α Γ k − η 1−α Γ k ∞ n n − Ψ n an zn−1 4.29 n with Ψ n Γ n−1 , Γ n−η ≤ η < 1, n 2, 3, , 4.30 reiθ < r < , then, for μ > and z 2π η Dz f z μ dθ ≤ 2π η Dz fk z μ dθ, 0≤η