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On a certain new subclass of meromorphic close-to-convex functions
Journal of Inequalities and Applications volume 2013, Article number: 164 (2013)
In this paper, we introduce and investigate a new subclass of meromorphic close-to-convex functions. For functions belonging to the class , we obtain some coefficient inequalities and a distortion theorem. The results presented here would unify and extend some recent work of Wang et al. (Appl. Math. Lett. 25:454-460, 2012).
Let Σ be the class of functions f of the form:
which are analytic in the punctured open unit disk .
Let P denote the class of functions p given by
which are analytic and convex in U and satisfy the condition ().
A function is said to be in the class of meromorphic starlike functions of order α if it satisfies the inequality
In addition, a function is said to be in the class MC of meromorphic close-to-convex functions if it satisfies the inequality
More recently, Wang et al.  discussed a class MK of meromorphic close-to-convex functions, that is, a function is said to be in the class MK if it satisfies the inequality
Let be analytic in U. If there exists a function , such that
For two functions f and g analytic in U, we say that the function is subordinate to in U, and we write () if there exists a Schwarz function , analytic in U with and , such that (). In particular, if the function g is univalent in U, then we have and (see, for example, ).
Motivated essentially by the above mentioned function classes MK and , we now introduce a new class of meromorphic functions.
Definition 1 Let denote the class of functions in Σ satisfying the inequality
where , , is a fixed positive integer and is defined by the following equality:
We note that (see ), so the class is a generation of the class MK.
In this paper, we prove that the class is a subclass of meromorphic close-to-convex functions. Moreover, we provide some coefficient inequalities and a distortion theorem for functions in the class . Our results unify and extend the corresponding results obtained by Wang et al. .
2 Main results
First of all, we give two meaningful conclusions about the class . The proof of Theorem 1 below is much akin to that of Theorem 1 in , so we choose to omit the details involved.
Theorem 1 A function if and only if there exists such that
From Theorem 1, we know that
because of ().
Lemma 1 Let , where (). Then for , we have
Proof Since (), by the definition of meromorphic starlike functions, we have
We now let
Differentiating (2.4) with respect to z logarithmically, we easily get
From (2.5) together with (2.3), we obtain
by noting that , which implies that
The proof of Lemma 1 is thus completed. □
Theorem 2 Let , then .
Proof From (1.4), we know
Since , by Lemma 1 and (2.6), we can easily get the assertion of Theorem 2. □
Remark 2 From Theorem 2 and the inequality (2.2), we see that if , then is a meromorphic close-to-convex function. So, is a subclass of the class MC of meromorphic close-to-convex functions.
Next, we give some coefficient inequalities for functions belonging to the class .
Theorem 3 Let and be analytic in . If for and , we have
where the coefficients () are given by (2.9), then .
By Theorem 2, we know that . Hence, equality (2.6) can be written as
To prove , it suffices to show that
where is given by (2.8). From (2.7), we know that
Now, by the maximum modulus principle, we deduce from (1.1), (2.9) and (2.10) that
This evidently completes the proof of Theorem 3. □
Theorem 4 Let . Then
where the coefficients () are given by (2.9).
Proof Suppose that . Then we know that
where is given by (2.8). After a simple computation, the inequality (2.2) is equivalent to
By substituting (1.1) and (2.9) into (2.13), we obtain the desired assertion (2.11) of Theorem 4.
Finally, we provide the following distortion theorem for the considered class of functions . □
Theorem 5 If , then
Proof If , then there exists a function such that (1.3) holds true. It follows from Theorem 2 that the function given by (2.8) is a meromorphic starlike function. Hence, we have (see )
Let us define by
Then, by using a similar method as in [, p.105], we have
Thus, from (2.15), (2.16) and (2.17), we readily get the inequality (2.14). The proof of Theorem 5 is thus completed. □
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The present investigation was partly supported by the Natural Science Foundation of China under Grant 11271045, the Higher School Doctoral Foundation of China under Grant 2010000311000 4 and the Natural Science Foundation of Inner Mongolia of China under Grant 2010MS0117. The authors are thankful to the referees for their careful reading and making some helpful comments which have essentially improved the presentation of this paper.
The authors declare that they have no competing interests.
All authors jointly worked on the results and they read and approved the final manuscript.
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Cite this article
Tang, H., Deng, GT. & Li, SH. On a certain new subclass of meromorphic close-to-convex functions. J Inequal Appl 2013, 164 (2013). https://doi.org/10.1186/1029-242X-2013-164
- analytic functions
- meromorphic close-to-convex functions
- coefficient inequality
- distortion theorem