Some Properties of Certain Class of Analytic Functions

We obtain some properties related to the coefficient bounds for certain subclass of analytic functions. We also work on the differential subordination for a certain class of functions. 1. Introductions Let denote the class of functions which is analytic in the unit disc . Let Now let be the class of functions defined by The Hadamard product of two functions and is defined by where and are analytic in . Let , , and then is analytic in the open unit disc . The function defined in (3) is equivalent to where is the Hadamard product and is analytic in the open unit disc . We introduce a class of functions where Authors like Saitoh [1] and Owa [2, 3] had previously studied the properties of the class of functions . They obtained many interesting results and Wang et al. [4] studied the extreme points, coefficient bounds, and radius of univalency of the same class of functions. They obtained the following theorem among other results. Theorem 1 (see [4]). Let . A function if and only if can be expressed as where is the probability measure defined on For fixed , , and , the class and the probability measure defined on are one-to-one by expression (8). Recently, Hayami et al. [5] studied the coefficient estimates of the class of function in the open unit disc . They derived results based on properties of the class of functions , . Xu et al. [6] used the principle of differential subordination and the Dziok-Srivastava convolution operator to investigate some analytic properties of certain subclass of analytic functions. We also note that Stanciu et al. [7] used the properties of the class of functions , , to investigate the analytic and univalent properties of the following integral operator: where . Motivated by the work in [1–7], we used the properties of the class of function , , to investigate the coefficient estimates of the class of functions in the open unit disc . We also use the principle of differential subordination to investigate some properties of the class of functions . We state the following known results required to prove our work. Definition 2. If and are analytic in , then is said to be subordinate to , written as or . If is univalent in , then and . Theorem 3 (see [8]). Consider if and only if there is probability measure on such that and . The correspondence between and the set of probability measures on given by Hallenbeck [9] is one-to-one. Theorem 4 (see [10, 11]). Let be convex in , , , and . If and then The function is convex and the best -dominant. Lemma 5 (see [10]). Let be starlike in , with and . If satisfies then and is the best