Some Applications of Second-Order Differential Subordination on a Class of Analytic Functions Defined by Komatu Integral Operator

We introduce a new class of analytic functions by using Komatu integral operator and obtain some subordination results. 1. Introduction, Definitions, and Preliminaries Let be the set of real numbers, the set of complex numbers, be the set of positive integers, and Let be the class of analytic functions in the open unit disk and the subclass of consisting of the functions of the form Let be the class of all functions of the form which are analytic in the open unit disk with Also let denote the subclass of consisting of functions which are univalent in . A function analytic in is said to be convex if it is univalent and is convex. Let denote the class of normalized convex functions in . If and are analytic in , then we say that is subordinate to , written symbolically as if there exists a Schwarz function which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence [1, page 4]: Let be a function and let 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 (13). A dominant , which satisfies for all dominants of (13), is said to be the best dominant of (13). Recently, Komatu [2] introduced a certain integral operator defined by Thus, if is of the form (5), then it is easily seen from (14) that (see [2]) Using the relation (15), it is easy verify that We note the following.(i)For and ( is any integer), the multiplier transformation was studied by Flett [3] and S？l？gean [4].(ii)For and ( ), the differential operator was studied by S？l？gean [4].(iii)For and ( is any integer), the operator was studied by Uralegaddi and Somanatha [5].(iv)For , the multiplier transformation was studied by Jung et al. [6]. Using the operator , we now introduce the following class. Definition 1. Let be the class of functions satisfying where , , and is the Komatu integral operator. In order to prove our main results, we will make use of the following lemmas. Lemma 2 (see [7]). 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 3 (see [8]). Let , , and let Let be an analytic function in with and suppose that If is analytic in and then where is a solution of the differential equation given by Moreover is the best dominant. 2. Main Results Theorem 4. The set is convex.