Neighborhoods and partial sums of certain subclass of starlike functions
© Wang et al.; licensee Springer. 2013
Received: 7 December 2012
Accepted: 8 March 2013
Published: 10 April 2013
The main purpose of the present paper is to derive the neighborhoods and partial sums of a certain subclass of starlike functions.
Keywordsanalytic functions starlike functions neighborhoods partial sums
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
By making use of the definition (2.1), we now derive the following result.
We thus complete the proof of Theorem 1. □
The bounds in (2.11) and (2.12) are sharp.
Combining (2.14) and (2.15), we deduce that the assertion (2.11) holds true.
which implies that the bound in (2.11) is the best possible for each .
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.
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 .
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.
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