Journal of Complex Analysis

Volume 2013, Article ID 512469, 3 pages

http://dx.doi.org/10.1155/2013/512469

## Certain Inequalities of Multivalent Analytic Functions with Missing Coefficients

^{1}Department of Mathematics, Suqian College, Suqian 223800, China^{2}Information 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., [1–7]). 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

- J.-L. Liu and H. M. Srivastava, “Some convolution conditions for starlikeness and convexity of meromorphically multivalent functions,”
*Applied Mathematics Letters*, vol. 16, no. 1, pp. 13–16, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Nunokawa, “On the order of strongly starlikeness of strongly convex functions,”
*Proceedings of the Japan Academy A*, vol. 69, no. 7, pp. 234–237, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Nunokawa, S. Owa, H. Saitoh, A. Ikeda, and N. Koike, “Some results for strongly starlike functions,”
*Journal of Mathematical Analysis and Applications*, vol. 212, no. 1, pp. 98–106, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Nunokawa and D. K. Thomas, “On convex and starlike functions in a sector,”
*Australian Mathematical Society A*, vol. 60, no. 3, pp. 363–368, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Obradović and S. Owa, “Some sufficient conditions for strongly starlikeness,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 24, no. 9, pp. 643–647, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Takahashi and M. Nunokawa, “A certain connection between starlike and convex functions,”
*Applied Mathematics Letters*, vol. 16, no. 5, pp. 653–655, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Xu and D.-G. Yang, “An application of differential subordinations and some criteria for starlikeness,”
*Indian Journal of Pure and Applied Mathematics*, vol. 36, no. 10, pp. 541–556, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D.-G. Yang and J.-L. Liu, “On a class of analytic functions with missing coefficients,”
*Applied Mathematics and Computation*, vol. 215, no. 9, pp. 3473–3481, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet