Neighborhoods and partial sums of certain subclass of starlike functions
Journal of Inequalities and Applications volume 2013, Article number: 163 (2013)
The main purpose of the present paper is to derive the neighborhoods and partial sums of a certain subclass of starlike functions.
Let denote the class of functions f of the form
which are analytic in the open unit disk
A function is said to be in the class of starlike functions of order β if it satisfies the inequality
Assuming that , and , we say that a function if it satisfies the condition
For convenience, throughout this paper, we write
Recently, Ravichandran et al.  proved that . Subsequently, Liu et al.  derived various properties and characteristics such as inclusion relationships, Hadamard products, coefficient estimates, distortion theorems and cover theorems for the class and a subclass of with negative coefficients. Furthermore, Singh et al.  generalized the class and found several sufficient conditions for starlikeness. In the present paper, we aim at proving the neighborhoods and partial sums of the class .
2 Main results
Following the earlier works (based upon the familiar concept of a neighborhood of analytic functions) by Goodman  and Ruscheweyh , and (more recently) by Altintaş et al. [6–9], Cǎtaş , Frasin , Keerthi et al.  and Srivastava et al. , we begin by introducing here the δ-neighborhood of a function of the form (1.1) by means of the definition
By making use of the definition (2.1), we now derive the following result.
Theorem 1 If satisfies the condition
Proof By noting that the condition (1.3) can be rewritten as follows:
we easily find from (2.4) that a function if and only if
which is equivalent to
It follows from (2.6) that
If satisfies the condition (2.2), we deduce from (2.5) that
We now suppose that
It follows from (2.1) that
Combining (2.7) and (2.8), we easily find that
which implies that
Therefore, we conclude that
We thus complete the proof of Theorem 1. □
Theorem 2 Let be given by (1.1) and define the partial sums of f by
The bounds in (2.11) and (2.12) are sharp.
Proof (1) Suppose that . We know that , which implies that
From (2.10), we easily find that
which implies that . In view of Theorem 1, we deduce that
(2) It is easy to verify that
Therefore, we have
We now suppose that
It follows from (2.13) and (2.14) that
which shows that
Combining (2.14) and (2.15), we deduce that the assertion (2.11) holds true.
Moreover, if we put
then for , we have
which implies that the bound in (2.11) is the best possible for each .
Similarly, we suppose that
In view of (2.13) and (2.17), we conclude that
which implies that
Combining (2.17) and (2.18), we readily get the assertion (2.12) of Theorem 2. The bound in (2.12) is sharp with the extremal function f given by (2.16).
The proof of Theorem 2 is thus completed. □
Finally, we turn to ratios involving derivatives. The proof of Theorem 3 below is much akin to that of Theorem 2, we here choose to omit the analogous details.
Theorem 3 Let be given by (1.1) and define the partial sums of f by (2.9). If the condition (2.10) holds, then
The bounds in (2.19) and (2.20) are sharp with the extremal function given by (2.16).
Remark By setting and in Theorems 2 and 3, we get the corresponding results obtained by Silverman .
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Dedicated to Professor Hari M Srivastava.
The present investigation was supported by the National Natural Science Foundation under Grants 11226088 and 11101053, the Key Project of Chinese Ministry of Education under Grant 211118, the Excellent Youth Foundation of Educational Committee of Hunan Province under Grant 10B002, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants 11FEFM02 and 12FEFM02, and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant 12A110002 of the People’s Republic of China.
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, ZG., Yuan, XS. & Shi, L. Neighborhoods and partial sums of certain subclass of starlike functions. J Inequal Appl 2013, 163 (2013). https://doi.org/10.1186/1029-242X-2013-163