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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 154912, 4 pages
http://dx.doi.org/10.1155/2013/154912
Research Article

Sufficient Conditions for Non-Bazilevič Functions

1School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan 455002, China
2School of Business, Central South University, Changsha, Hunan 410083, China

Received 11 July 2013; Accepted 19 September 2013

Academic Editor: Alberto Fiorenza

Copyright © 2013 Qun-Fa Cui et al. 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 main purpose of this paper is to derive some sufficient conditions for analytic functions to be of non-Bazilevič type.

1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disk:

For and , a function is said to be in the class if it satisfies the condition As usual, the class is said to be non-Bazilevič functions of order (see [1]).

For some recent investigations of non-Bazilevič functions, see, for example the works of [26] and the references cited therein.

For two functions and , analytic in , we say that the function is subordinate to in and write if there exists a Schwarz function , which is analytic in with such that Indeed, it is known that Furthermore, if the function is univalent in , then we have the following equivalence:

To derive our main results, we need the following lemmas.

Lemma 1 (see [7]). Let be analytic in and let be analytic and starlike (with respect to the origin) univalent in with . If then

Lemma 2 (see [8]). Let be univalent in . Also let be analytic in the domain containing with when . Set Suppose that (1) is starlike univalent in ;(2) for .
If is analytic in with , and then , and is the best dominant.

Lemma 3 (see [9]). Let be a set in the complex plane and suppose that is a mapping from to which satisfies for and for all real such that .
If the function is analytic in and for all , then .

In this paper, we aim at proving some sufficient conditions for analytic functions to be of non-Bazilevič type.

2. Main Results

Our first main result is given by Theorem 4.

Theorem 4. Suppose that is starlike in with . If then

Proof. We define the function by Then is analytic in with . It follows from (15) that Combining (13) and (16), we find that By Lemma 1, we deduce that From (15) and (18), we readily get the assertion (14) of Theorem 4.

Theorem 5. If satisfies the inequality then .

Proof. Suppose that the function is defined by (15). It follows that Combining (19) and (20), we know that An application of Lemma 1 to (21) yields By noting that which implies that the region is symmetric with respect to the real axis and is convex univalent in therefore, we have
Combining (15), (22), and (24), we conclude that This completes the proof of Theorem 5.

Theorem 6. Suppose that is convex in with . If then and is the best dominant.

Proof. Suppose that the function is defined by (15). It follows that
We now assume that Obviously, and are analytic in the plane. By noting that the function is starlike in and it follows from (26) that Combining (27), (29), and Lemma 2, we get the assertion of Theorem 6.

Remark 7. By taking suitable and in Theorems 4 and 6, respectively, we can get some useful consequences. Here we choose to omit the details.

Theorem 8. If satisfies the condition then .

Proof. Suppose that Then is analytic in . It follows from (35) that where For all real and satisfying , we have
We now put Then for all real such that . Moreover, in view of (34), we know that . Thus, by Lemma 3, we deduce that which shows that the desired assertion of Theorem 8 holds.

Acknowledgments

The present investigation was supported by the National Natural Science Foundation under Grant nos. 11301008, 11226088, 71171024, 71371195, and 70921001 and the Key Project of Natural Science Foundation of Educational Committee of Henan Province under Grant no. 12A110002 of the People’s Republic of China.

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