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

Serap Bulut Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Kartepe-Kocaeli, Turkey Correspondence should be addressed to Serap Bulut; bulutserap@yahoo.com Received 24 December 2013; Accepted 13 February 2014; Published 12 March 2014 Academic Editors: G. Mantica, C. Mascia, A. Peris, and W. Shen Copyright © 2014 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. We introduce a new class of analytic functions by using Komatu integral operator and obtain some subordination results.

N := {1, 2, 3, . ..} = N 0 \ {0} (1) be the set of positive integers, and Let H be the class of analytic functions in the open unit disk and H[, ] the subclass of H consisting of the functions of the form Let A  be the class of all functions of the form which are analytic in the open unit disk U with Also let S denote the subclass of A consisting of functions  which are univalent in U.
A function  analytic in U is said to be convex if it is univalent and (U) is convex. Let denote the class of normalized convex functions in U.
If  and  are analytic in U, then we say that  is subordinate to , written symbolically as if there exists a Schwarz function  which is analytic in U with Indeed, it is known that Furthermore, if the function  is univalent in U, then we have the following equivalence [1, page 4]:

ISRN Mathematical Analysis
Let  : C 3 × U → C be a function and let ℎ be univalent in U.If  is analytic in U and satisfies the (second-order) differential subordination  ( () ,   () ,  2   () ; ) ≺ ℎ () , ( ∈ U) , 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 q, which satisfies q ≺  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  ∈ A 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.
Using the operator    , we now introduce the following class.
In order to prove our main results, we will make use of the following lemmas.
Lemma 2 (see [7]).Let ℎ be a convex function with ℎ(0) =  and let  ∈ C * := C −{0} be a complex number with R{} ≥ 0. If  ∈ H[, ] and then where The function  is convex and is the best dominant.
Lemma 3 (see [8]).Let R{} > 0,  ∈ N, and let Let ℎ be an analytic function in U with ℎ(0) = 1 and suppose that If is analytic in U and then where  is a solution of the differential equation given by Moreover  is the best dominant.

Main Results
For any nonnegative numbers  1 ,  2 , . . .,   such that we must show that the function By ( 28) and (31), we have Therefore we get Differentiating (34) with respect to , we obtain So we get since  1 +  2 + ⋅ ⋅ ⋅ +   = 1.Therefore we get the desired result.
Theorem 5. Let  be convex function in U with (0) = 1 and let where  is a complex number with R{} > −2.If  ∈ R , () and F = I  , where then implies and this result is sharp.