On a certain new subclass of meromorphic close-to-convex functions
© Tang et al.; licensee Springer. 2013
Received: 23 September 2012
Accepted: 14 March 2013
Published: 10 April 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).
which are analytic in the punctured open unit disk .
which are analytic and convex in U and satisfy the condition ().
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.
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.
because of ().
The proof of Lemma 1 is thus completed. □
Theorem 2 Let , then .
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 .
where the coefficients () are given by (2.9), then .
This evidently completes the proof of Theorem 3. □
where the coefficients () are given by (2.9).
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 . □
Thus, from (2.15), (2.16) and (2.17), we readily get the inequality (2.14). The proof of Theorem 5 is thus completed. □
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.
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