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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 391038, 8 pages

http://dx.doi.org/10.1155/2012/391038

## Argument Property for Certain Analytic Functions

Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Received 18 September 2011; Accepted 1 November 2011

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Qing Yang and Jin-Lin Liu. 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

Let *P* be the class of functions *p*(*z*) of the form which are analytic in the open unit disk . The object of the present paper is to derive certain argument inequalities of analytic functions *p*(*z*) in *P*.

#### 1. Introduction

Let be the class of functions of the form which are analytic in the open unit disk . For functions and in the class , we say that is subordinate to if there exists an analytic function in with , , and such that . We denote this subordination by If is univalent in , then this subordination is equivalent to and .

Recently, several authors investigated various argument properties of analytic functions (see, e.g., [1–6]). The object of the present paper is to discuss some argument inequalities for in the class .

Throughout this paper, we let

In order to prove our main result, we well need the following lemma.

Lemma 1.1 (see [6]). *Let and . Also let
**
If satisfies
**
where
**
is (close to convex) univalent, then
**
The bounds and in (1.7) are sharp for the function defined by
*

*Remark 1.2 (see [6]). *The function defined by (1.8) is analytic and univalently convex in and

#### 2. Main Result

Our main theorem is given by the following.

Theorem 2.1. *Let
**
If satisfies
**
where
**
where denotes , then
**
The bounds and in (2.2) are the largest numbers such that (2.4) holds true. *

*Proof. *By taking in Lemma 1.1, we find that if satisfies
where
then (2.4) holds true.

For , and , we get

We consider the following two cases.

(i) If
then from (2.7), and (2.6), we have
and so
where ,, , ,

We now calculate the maximum value of . It is easy to verify that
and that

Set
then . Noting that
we easily have
Hence, , and it follows from (2.11) to (2.16) that
where denotes . Thus, by using (2.1), (2.10), and (2.17), we arrive at

(ii) If , then we obtain
which leads to
where ,, , ,

Now, we have
Let
then , ,
Hence, we deduce that and
where . Further, by using (2.1), (2.20), and (2.25), we find that
In view of , we conclude from (2.18) and (2.26) that properly contains the angular region in the complex -plane. Therefore, if satisfies (2.2), then the subordination relation (2.5) holds true, and thus we arrive at (2.4).

Furthermore, for the function defined by (1.8), we have
Hence, by using (2.18) and (2.25), we see that the bounds and in (2.2) are best possible.

#### Acknowledgment

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

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