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Journal of Complex Analysis
Volume 2013 (2013), Article ID 512469, 3 pages
http://dx.doi.org/10.1155/2013/512469
Research Article

Certain Inequalities of Multivalent Analytic Functions with Missing Coefficients

1Department of Mathematics, Suqian College, Suqian 223800, China
2Information Engineering College, Yangzhou University, Yangzhou 225009, China

Received 22 November 2012; Accepted 31 December 2012

Academic Editor: Jacek Dziok

Copyright © 2013 Yi-Ling Cang and Cai-Mei Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of the present paper is to derive the radius of starlikeness for certain p-valent functions with missing coefficients. The results obtained here are sharp.

1. Introduction

Let denote the class of functions of the form which are -valent analytic in the open unit disk . A function is said to be -valently starlike of order in if it satisfies

For functions and analytic in , we say that is subordinate to in , and we write , if there exists an analytic function in such that Furthermore, if the function is univalent in , then

A number of results for -valently starlike functions have been obtained by several authors (see, e.g., [17]). In this note, we shall derive the radius of starlikeness for certain -valent functions with missing coefficients.

2. Main Results

Our main result is the following.

Theorem 1. Let belong to the class and satisfy Then, where , and is the smallest root in of the equation The result is sharp.

Proof. From (5), we can write that where is analytic and in . Differentiating both sides of (8) logarithmically, we arrive at Put and   . Then, (8) implies that With the help of the Carathéodory inequality: it follows from (10) that where , and From (10), we can see that Thus, we have from (10), (11), and (15) that Since is a even function of , from (13), (14), and (15), we see that
Let us now calculate the minimum value of on the closed interval . Noting that (see [8]) and lower bound in (11), we deduce from (18) that where Also, . Suppose that . Then, Hence, by virtue of the mathematical induction, we have for all and . This implies that In view of (20) and (24), we see that when , then Further, it follows from (13), (18), and (25) that where and Note that and . If we let denote the smallest root in of the equation , then (26) yields the desired result (6).
To see that the bound is the best possible one, we consider the function It is clear that for , which shows that the bound cannot be increased.
Setting , Theorem 1 reduces to the following result which gives the radius of starlikeness for certain -valent analytic functions with missing coefficients.

Corollary 2. Let satisfy the condition (5) and . Then, is starlike of order in The result is sharp.

Acknowledgment

The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped them to improve the paper.

References

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