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Some basic properties of certain subclasses of meromorphically starlike functions
Journal of Inequalities and Applications volume 2014, Article number: 29 (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.
Let Σ denote the class of functions f of the form
which are analytic in the punctured open unit disk
A function is said to be in the class of meromorphically starlike functions of order α if it satisfies the inequality
Let denote the class of functions p given by
which are analytic in and satisfy the condition
Given two functions , where f is given by (1.1) and g is given by
the Hadamard product (or convolution) is defined by
A function is said to be in the class if it satisfies the condition
where (and throughout this paper unless otherwise mentioned) the parameters β and λ are constrained as follows:
Clearly, we have
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.
Let denote the subset of such that all functions having the following form:
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 )
If the function is given by (1.2), then
Lemma 2.2 Let and . Suppose also that the sequence is defined by
Proof By virtue of (2.1), we easily get
Combining (2.3) and (2.4), we obtain
Thus, for , we deduce from (2.5) that
The proof of Lemma 2.2 is evidently completed. □
Lemma 2.3 Let
Suppose also that is given by (1.1). If
where (and throughout this paper unless otherwise mentioned) the parameter γ is constrained as follows:
Lemma 2.4 Let be given by (1.5). Suppose also that γ is defined by (2.8) and the condition (2.6) holds true. Then if and only if
3 Main results
We begin by proving the following coefficient estimates for functions belonging to the class .
Theorem 3.1 Let γ be defined by (2.8). If with , then
Proof Suppose that
Then, by the definition of the function class , we know that q is analytic in and
It follows from (2.8) and (3.1) that
By noting that
if we put
by Lemma 2.1, we know that
It follows from (3.2) that
In view of (3.3), we get
From (3.4), we obtain
Moreover, we deduce from (3.5) that
Next, we define the sequence as follows:
In order to prove that
we make use of the principle of mathematical induction. By noting that
Therefore, assuming that
Combining (3.7) and (3.8), we get
Hence, by the principle of mathematical induction, we have
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. □
Following the earlier works (based upon the familiar concept of neighborhood of analytic functions) by Goodman  and Ruscheweyh , and (more recently) by Altintaş et al. [21–24], Cǎtaş , Cho et al. , Liu and Srivastava [27–29], Frasin , Keerthi et al. , Srivastava et al.  and Wang et al. . Assuming that γ is given by (2.8) and the condition (2.6) of Lemma 2.3 holds true, we here introduce the δ-neighborhood of a function of the form (1.1) by means of the following definition:
By making use of the definition (3.10), we now derive the following result.
Theorem 3.2 Let the condition (2.6) hold true. If satisfies the condition
Proof By noting that the condition (1.3) can be written as
we easily find from (3.13) that a function if and only if
which is equivalent to
It follows from (3.15) that
If given by (1.1) satisfies the condition (3.11), we deduce from (3.14) that
We now suppose that
It follows from (3.10) that
Combining (3.16) and (3.17), we easily find that
which implies that
Therefore, we have
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.
Theorem 3.3 Let be given by (1.1) and define the partial sums of f by
where γ is given by (2.8) and the condition (2.6) holds true, then
The bounds in (3.20) and (3.21) are sharp.
Proof First of all, we suppose that
We know that
From (3.19), we easily find that
which implies that . By virtue of Theorem 3.2, we deduce that
Next, it is easy to see that
Therefore, we have
We now suppose that
It follows from (3.22) and (3.23) that
which shows that
Combining (3.23) and (3.24), we deduce that the assertion (3.20) holds true.
Furthermore, if we put
which implies that the bound in (3.20) is the best possible for each .
Similarly, we suppose that
In view of (3.22) and (3.26), we conclude that
which implies that
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.
Theorem 3.4 Let be given by (1.1) and define the partial sums of f by (3.18). If the conditions (2.6) and (3.19) hold, where γ is given by (2.8), then
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 .
Theorem 3.5 Let
Proof Suppose that . Then
Since and , we find that
It follows from (3.31) and (3.32) that
which shows that , and subsequently, we see that , that is,
Now, by setting
we easily find from (3.33) and (3.34) that
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.
Corollary 3.6 Let
By virtue of Lemma 2.4, we obtain the following result.
Corollary 3.7 Let . Suppose also that γ is defined by (2.8) and the condition (2.6) holds true. Then
Each of these inequalities is sharp, with the extremal function given by
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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.
The authors declare that they have no competing interests.
The authors completed the paper together. They also read and approved the final manuscript.
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Wang, Z., Srivastava, H. & Yuan, S. Some basic properties of certain subclasses of meromorphically starlike functions. J Inequal Appl 2014, 29 (2014). https://doi.org/10.1186/1029-242X-2014-29
- meromorphic function
- starlike function
- Hadamard product (or convolution)
- partial sum