Abstract
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.
1. Introduction and Preliminaries
Let denote the class of functions of the form which are analytic in the punctured open unit disk
For two functions and , analytic in , we say that the function is subordinate to in and write 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:
Let , where is given by (1) and is defined by Then the Hadamard product (or convolution) of the functions and is defined by
A function is said to be in the class of meromorphic starlike functions if it satisfies the inequality A function is said to be in the class of meromorphic convex functions if it satisfies the inequality Moreover, a function is said to be in the class of meromorphic close-to-convex functions if it satisfies the condition
Let be analytic in . If there exists a function such that then we say that , where denotes the usual class of convex functions. The function class 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 , we now introduce and investigate the following class of meromorphic close-to-convex functions.
Definition 1. A function is said to be in the class 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 if and only if
We observe that
and, thus, the function class is a subclass of meromorphic close-to-convex functions.
Clearly, the class is the familiar class of meromorphic close-to-convex functions of order .
For some recent investigations of meromorphic functions, see, for example, the works of [7–22] and the references cited therein.
To derive our main results, we need the following lemmas.
Lemma 2 (see [23]). Let be analytic in and let be analytic and convex in . If , then
Lemma 3 (see [24]). Suppose that Then Each of these inequalities is sharp, with the extremal function given by
Lemma 4 (see [25]). Let and . Then, if and only if
Lemma 5 (see [26]). Suppose that the function . Then
Lemma 6 (see [27]). Suppose that Then, where is the unique root of the equation in the interval . 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 .
2. Main Results
We begin by stating the following coefficient inequality of the class .
Theorem 7. Suppose that Then
Proof. Let 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 , it is well known that .
On the other hand, we find from (38) that
By noting that , it follows from Lemma 3 that
Combining (37), (39), and (40), we have
Thus, the assertion (33) of Theorem 7 follows directly from (41).
Theorem 8. Let
If satisfies the condition
then .
Proof. To prove , 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.
Example 9. By applying Theorem 8, it is obvious to see that the function
Theorem 10. Let and . A function if and only if
Proof. A function if and only if 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.
Theorem 11. Let Then,
Proof. Suppose that . We easily know that By setting , , , and , it follows from (51) that In view of Lemma 4, we deduce that which implies that . Thus, the assertion (52) of Theorem 11 holds.
Theorem 12. Let . Then,
Proof. Let . By definition, we know that Suppose that the function is defined by (36). Then, we have Since , by Lemma 5, we know that Thus, by virtue of (36), (58), and (59), we readily get the assertion (56) of Theorem 12.
Finally, we derive the radius of meromorphic convexity for the class .
Theorem 13. Let with . Then,(1)for , is meromorphic convex in ;(2)for , is meromorphic convex in ,where is the unique root of the equation in the interval and and are the smallest root of the equations in the interval , respectively.
Proof. Let and suppose that
Then,
It follows from (62) that
Differentiating both sides of (64) logarithmically, we get
Since , we know that
Combining (63), (65), (66), and Lemma 6, we obtain
where is the unique root of (30) in the interval . It follows from (67) that the bound of meromorphic convexity for the class is determined either by the equation
or by the equation
We note that (68) and (69) can be rewritten as follows:
Let and be the smallest root of the equations and in the interval , respectively. By observing that , we deduce that for .
Similarly, we know that for , since .
We observe that
Thus, when , is meromorphic convex in ; when , is meromorphic convex in .
The proof of Theorem 13 is thus completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation under Grant nos. 11301008 and 11226088, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant no. 14B110012 of the People's Republic of China. The authors would like to thank the referees for their valuable comments and suggestions which essentially improved the quality of this paper.