Abstract and Applied Analysis

Volume 2014 (2014), Article ID 634710, 4 pages

http://dx.doi.org/10.1155/2014/634710

Research Article

Applications of Differential Subordination for Argument Estimates of Multivalent Analytic Functions

Meng-Ting Lu, Ting Jia, Xing-Qian Ling, and Jin-Lin Liu

Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Received 21 November 2013; Accepted 20 January 2014; Published 26 February 2014

Academic Editor: David Kalaj

Copyright © 2014 Meng-Ting Lu 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.

By using the method of differential subordinations, we derive some properties of multivalent analytic functions. All results presented here are sharp.

This paper is dedicated to Professor Miodrag Mateljević on the occasion of his 65th birthday

1. Introduction

Let denote the class of functions of the form

which are analytic in the open unit disk . Let and be analytic in . Then, we say that is subordinate to in , written as , if there exists an analytic function in , such that and . If is univalent in , then the subordination is equivalent to and . Let be analytic in . Then, for , it is clear that

if and only if

Recently, a number of results for argument properties of analytic functions have been obtained by several authors (see, e.g., [1–5]). The objective of the present paper is to derive some further interesting properties of multivalent analytic functions. The basic tool used here is the method of differential subordinations.

To derive our results, we need the following lemmas.

Lemma 1 (see [6, Theorem 1, page 776]). Let be analytic and starlike univalent in with . If is analytic in and , then

Lemma 2 (see [5, Theorem 1, page 1814]). Let , , , and . Also let

If is analytic in with and

where

is (close-to-convex) univalent in , then

The bounds and in (9) are sharp for the function defined by

Remark 3 (see [5, Lemma 2, page 1813]). The function defined by (10) is analytic and univalent convex in and

2. Main Results

Our first result is contained in the following.

Theorem 4. Let and . If satisfies and

where is the smallest positive root of the equation

then

The bound is sharp for each .

Proof. Let

We can see easily that (13) has two positive roots. Since and , we have

Put

Then, from the assumption of the theorem, we can see that is analytic in with and for all . Taking the logarithmic differentiations in both sides of (17), we get

for all . Thus, inequality (12) is equivalent to

By using Lemma 1, (20) leads to

or to

According to (16), (22) can be written as

Now, by taking and in (2) and (3), we have

for all because of . This proves (14).

Next, we consider the function defined by

for all . It is easy to see that

for all . Since

it follows from (3) that

Hence, we conclude that the bound is the best possible for each .

Next, we derive the following.

Theorem 5. If satisfies and

where

then

The bound in (31) is sharp.

Proof. Let

Then, from the assumption of the theorem we can see that is analytic in with and for all . According to (32) and (29), we have immediately

that is,

Now, by using Lemma 1, we obtain

Since the function is convex univalent in and

from (35), we get inequality (31).

To show that the bound in (31) cannot be increased, we consider

It is easy to verify that the function satisfies inequality (29). On the other hand, we have

as . Now, the proof of the theorem is complete.

Finally, we discuss the following theorem.

Theorem 6. Let . If satisfies and

for all , where

then

The bound in (39) is sharp.

Proof. Define the function by (17). For , it follows from (17) and (18) that

for all . Putting

in Lemma 2 and using (42), we see that if

where

then (41) holds true.

Letting and , we deduce that

Making use of (46), we obtain that

Therefore, if satisfies (39), then the subordination (44) holds, and, thus, we obtain (41).

For the function

we find that

where is defined by (45). In view of (46) and (49), we conclude that the bound in (39) is the largest number such that (41) holds true. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

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

References

A. Gangadharan, V. Ravichandran, and T. N. Shanmugam, “Radii of convexity and strong starlikeness for some classes of analytic functions,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 301–313, 1997.

J.-L. Liu, “The Noor integral and strongly starlike functions,” Journal of Mathematical Analysis and Applications, vol. 261, no. 2, pp. 441–447, 2001.

M. Nunokawa, S. Owa, E. Y. Duman, and M. Aydoǧan, “Some properties of analytic functions relating to the Miller and Mocanu result,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1291–1295, 2011.

N.-E. Xu, D.-G. Yang, and S. Owa, “On strongly starlike multivalent functions of order β and type α,” Mathematische Nachrichten, vol. 283, no. 8, pp. 1207–1218, 2010.

D.-G. Yang and J.-L. Liu, “Argument inequalities for certain analytic functions,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1812–1821, 2010.

T. J. Suffridge, “Some remarks on convex maps of the unit disk,” Duke Mathematical Journal, vol. 37, no. 4, pp. 775–777, 1970.