Differential Subordination with Generalized Derivative Operator of Analytic Functions

Motivated by generalized derivative operator defined by the authors (El-Yagubi and Darus, 2013) and the technique of differential subordination, several interesting properties of the operator are given. 1. Introduction Let denote the class of functions of the form which are analytic in the open unit disk . Also let be the the subclass of consisting of all functions which are univalent in . We denote by and the familiar subclasses of consisting of functions which are, respectively, starlike of order and convex of order in :. Let be the class of holomorphic function in unit disk . For and we let Let two functions given by and be analytic in . Then the Hadamard product (or convolution) of the two functions , is defined by Recall that the function is subordinate to if there exists the Schwarz function , analytic in , with and such that , . We denote this subordination by . If is univalent in , then the subordination is equivalent to and . Let and be univalent in . If is analytic in and satisfies the (second order) differential subordination then is called a solution of the differential subordination. The univalent function is called a dominant of the solutions of the differential subordination, or more simply a dominant, if for all satisfying (5). A dominant that satisfies for all dominants of (5) is said to be the best dominant of (5) (note that the best dominant is unique up to a rotation of ). In order to prove the original results we need the following lemmas. Lemma 1 (see [1]). Let be a convex function with and let be a complex number with . If and then where The function is convex and is the best dominant. Lemma 2 (see [2]). Let be a convex function in and let where and is a positive integer. If is analytic in and then and this result is sharp. Lemma 3 (see [3]). Let ; if then belongs to the class of convex functions. We now state the following generalized derivative operator [4]: where , for , and is the Pochhammer symbol defined by Here can also be written in terms of convolution as To prove our results, we need the following inclusion relation: where is analytic function given by . 2. Main Results In the present paper, we will use the method of differential subordination to derive certain properties of generalised derivative operator . Note that differential subordination has been studied by various authors, and here we follow similar works done by Oros [5] and G. Oros and G. I. Oros [6]. Definition 4. For , and , let denote the class of functions which satisfy the condition Also, let denote the class of functions which satisfy the condition Remark