On a Subclass of Meromorphic Close-to-Convex Functions

The main purpose of this paper is to introduce and investigate a certain subclass of meromorphic close-to-convex functions. Such results as coefficient inequalities, convolution property, inclusion relationship, distortion property, and radius of meromorphic convexity are derived.


Introduction and Preliminaries
Let Σ denote the class of functions of the form For two functions and , analytic in U, we say that the function is subordinate to in U and write if there exists a Schwarz function , which is analytic in U with such that ( ) = ( ( )) ( ∈ U) .
Let , ∈ Σ, where is given by (1) and is defined by Then the Hadamard product (or convolution) * of the functions and is defined by A function ∈ Σ is said to be in the class MS * of meromorphic starlike functions if it satisfies the inequality A function ∈ Σ is said to be in the class MK of meromorphic convex functions if it satisfies the inequality 2 The Scientific World Journal Moreover, a function ∈ Σ is said to be in the class MC of meromorphic close-to-convex functions if it satisfies the condition R ( ( ) ( ) ) < 0 ( ∈ U; ∈ MS * ) . Let be analytic in U. If there exists a function g ∈ K such that then we say that f ∈ C( , ), where K denotes the usual class of convex functions. The function class C( , ) was introduced and studied recently by Peng [1] (see also Peng and Han [2], Selvaraj [3], Gao and Zhou [4], Kowalczyk and Leś-Bomba [5], and Xu et al. [6]). Motivated essentially by the above mentioned function class C( , ), we now introduce and investigate the following class of meromorphic close-to-convex functions.

Definition 1. A function
∈ Σ is said to be in the class MC( , ) if it satisfies the inequality where (and throughout this paper unless otherwise mentioned) the parameters and are constrained as follows: It is easy to verify that ∈ MC( , ) if and only if We observe that and, thus, the function class MC( , ) is a subclass of meromorphic close-to-convex functions. Clearly, the class MC( , 1) =: MC( ) is the familiar class of meromorphic close-to-convex functions of order .
To derive our main results, we need the following lemmas.
Lemma 2 (see [23]). Let be analytic in U and let be analytic and convex in U.
Lemma 3 (see [24]). Suppose that Then Each of these inequalities is sharp, with the extremal function given by Lemma 4 (see [25]). Let −1 ≦ < ≦ 1 and −1 ≦ < ≦ 1. Then, if and only if Lemma 5 (see [26]). Suppose that the function g ∈ MS * . Then Lemma 6 (see [27]). Suppose that Then, The Scientific World Journal 3 where 0 is the unique root of the equation in the interval (0, 1). The results are sharp.
In the present paper, we aim at proving some coefficient inequalities, convolution property, inclusion relationship, distortion property, and radius of meromorphic convexity of the class MC( , ).

Main Results
We begin by stating the following coefficient inequality of the class MC( , ).

Theorem 7. Suppose that
Proof. Let ∈ MC( , ) and suppose that where It follows that In view of Lemma 2, we know that By substituting the series expressions of functions , , and into (34), we get Since is univalent in U * , it is well known that | 1 | ≦ 1.
Proof. To prove ∈ MC( , ), it suffices to show that (15) holds. From (43), we know that Now, by the maximum modulus principle, we deduce from (1) and (44) that This evidently completes the proof of Theorem 8.
It is easy to see that condition (48) can be written as We observe that By substituting (50) into (49), we get the desired assertion (47) of Theorem 10.
Finally, we derive the radius of meromorphic convexity for the class MC( ).