Some basic properties of certain subclasses of meromorphically starlike functions
© Wang et al.; licensee Springer. 2014
Received: 9 October 2013
Accepted: 5 January 2014
Published: 24 January 2014
In this paper, we introduce and investigate certain subclasses of meromorphically starlike functions. Such results as coefficient inequalities, neighborhoods, partial sums, and inclusion relationships are derived. Relevant connections of the results derived here with those in earlier works are also pointed out.
Keywordsmeromorphic function starlike function Hadamard product (or convolution) neighborhood partial sum
In a recent paper, Wang et al.  had proved that if , then , which implies that the class is a subclass of the class of meromorphically starlike functions of order λ.
In the present paper, we aim at proving some coefficient inequalities, neighborhoods, partial sums and inclusion relationships for the function classes and .
2 Preliminary results
In order to prove our main results, we need the following lemmas.
Lemma 2.1 (See )
The proof of Lemma 2.2 is evidently completed. □
3 Main results
We begin by proving the following coefficient estimates for functions belonging to the class .
By means of Lemma 2.2 and (3.8), we know that (2.2) holds true. Combining (3.9) and (2.2), we readily get the coefficient estimates asserted by Theorem 3.1. □
By making use of the definition (3.10), we now derive the following result.
The proof of Theorem 3.2 is thus completed. □
Next, we derive the partial sums of the class . For some recent investigations involving the partial sums in analytic function theory, one can find in [28, 29, 34, 35] and the references cited therein.
The bounds in (3.20) and (3.21) are sharp.
Combining (3.23) and (3.24), we deduce that the assertion (3.20) holds true.
which implies that the bound in (3.20) is the best possible for each .
Combining (3.26) and (3.27), we readily get the assertion (3.21) of Theorem 3.3. The bound in (3.21) is sharp with the extremal function f given by (3.25). We thus complete the proof of Theorem 3.3. □
In what follows, we turn to quotients involving derivatives. The proof of Theorem 3.4 below is similar to that of Theorem 3.3, we here choose to omit the analogous details.
The bounds in (3.28) and (3.29) are sharp with the extremal function given by (3.25).
Finally, we prove the following inclusion relationship for the function class .
Therefore, the assertion (3.30) of Theorem 3.5 holds true. □
From Theorem 3.5 and the definition of the function class , we easily get the following inclusion relationship.
By virtue of Lemma 2.4, we obtain the following result.
The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008, 11226088, 11301041 and 11101053, 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.
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