Research Article | Open Access
On Certain New Subclass of Meromorphic Close-to-Convex Functions
We introduce a certain new subclass of meromorphic close-to-convex functions. Such results as inclusion relationship, coefficient inequalities, distortion, and growth theorems for this class of functions are derived.
Let denote the class of functions of the form which are analytic in the punctured open unit disk:
Let denote the class of functions given by which are analytic and convex in and satisfy the following condition:
A function is said to be in the class of meromorphic starlike functions of order if it satisfies the following inequality: Moreover, a function is said to be in the class of meromorphic close-to-convex functions if it satisfies the following condition:
For two functions and analytic in , we say that the function is subordinate to in and write , if there exists a Schwarz function , analytic in with and such that . Indeed, it is well known that Furthermore, if the function is univalent in , then we have the following equivalence:
Recently, Wang et al.  introduced and investigated the class of meromorphic close-to-convex functions which satisfy the inequality where . Kowalczyk and Leś-Bomba  discussed the class of analytic functions related to the starlike functions; a function which is analytic in is said to be in the class , if it is satisfies the following inequality: where . Şeker  discussed the class of analytic functions which satisfy the following condition: where , and .
Motivated essentially by the classes , and , we introduce and study the following more generalized class of meromorphic functions.
Definition 1. A function is said to be in the class if it satisfies the following inequality: where is a fixed positive integer, and is given by
We observe that the inequality (12) is equivalent to
Since , the class is a generalization of the class .
For some recent investigations on the class of close-to-convex functions, one can find them in [4–7] and the references cited therein. In the present paper, we aim at proving that the class is a subclass of meromorphic close-to-convex functions. Furthermore, some interesting results of the class are derived.
2. Preliminary Results
To prove our main results, we need the following lemmas.
Lemma 2. Let , where . Then for , one has
Lemma 3 (see ). Let . Then
Lemma 4 (see ). Suppose that . Then
Lemma 5 (see [10, page 105]). If the function analytic and convex in and satisfies the condition then
Lemma 7 (see ). Suppose that Then Each of these inequalities is sharp, with the extremal function given by
Lemma 8 (see ). Let be analytic in and let be analytic and convex in . If , then
Lemma 9. If , where is given by (3), then
3. Main Results
We first give the following result.
Theorem 10. Let , Then where is given by (13).
Now, we prove a sufficient condition for functions to belong to the class .
Theorem 12. Let . If for , one has where the coefficients () are given by (34), then .
Next, we give the inclusion relationship for class .
Theorem 13. Let . Then one has
In what follows, we derive the coefficient inequality for the class .
Theorem 14. Suppose that Then
Proof. Suppose that ; we know that where is given by (34). If we set it follows that In view of Definition 1 and Lemma 9, we know that By substituting the series expressions of functions , and into (49), we obtain Since is univalent in , it is well known that On the other hand, we find from (52) that Combining (28), (51), and (54), we have Thus, the assertion (47) of Theorem 14 follows directly from (55).
Finally, we give the distortion and growth theorems for the function class .
Theorem 15. If , then
Proof. If , then there exists a function such that (12) holds. It follows from Theorem 10 that the function given by (34) is a meromorphic starlike function. Therefore, by Lemma 4, we have Let be defined by (49); by Lemma 5, we know that Thus, from (49), (58), and (59), we readily get (56). Upon integrating (56) from to , we get (57). The proof of Theorem 15 is thus completed.
The present investigation was supported by the National Natural Science Foundation under Grant 11226088, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002, and the Science and Technology Program of Educational Department of Jiangxi Province under Grant GJJ12322 of China. The authors are grateful to the referees for their valuable comments and suggestions which essentially improved the quality of the paper.
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