Journal of Function Spaces
Volume 2016 (2016), Article ID 1782916, 4 pages
http://dx.doi.org/10.1155/2016/1782916

Research Article

On Subordinations for Certain Multivalent Analytic Functions in the Right-Half Plane

Yi-Hui Xu¹ and Jin-Lin Liu²

¹Department of Mathematics, Suqian College, Suqian 223800, China
²Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Received 2 January 2016; Accepted 16 February 2016

Academic Editor: Gestur Ólafsson

Copyright © 2016 Yi-Hui Xu 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.

The object of the present paper is to investigate some properties of multivalent analytic functions associated with the lemniscate of Bernoulli.

1. Introduction

Let denote the class of functions of the form

which are analytic in the open unit disk . A function is said to be starlike of order in if it satisfies

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

Furthermore, if the function is univalent in , then

Let be the class of functions defined by

Thus a function if lies in the region bounded by the right-half of the lemniscate of Bernoulli given by . In terms of subordination, the class consists of normalized analytic functions satisfying . This class was introduced by Sokól and Stankiewicz [1]. Several geometric properties had been investigated by Sokól and Stankiewicz [1]. They determined the radius of convexity for functions in the class . They also obtained structural formula, as well as growth and distortion theorems for these functions. Estimates for the first few coefficients of functions in were obtained in [2]. Recently, Sokól [3] determined various radii for functions belonging to the class ; these include the radii of convexity, starlikeness, and strong starlikeness of order . The -radii for certain well-known classes of functions including the Janowski starlike functions were obtained in [4]. General radii problems were also recently considered in [5, 6] wherein certain radii results for the class were obtained as special cases.

Motivated essentially by the above and some recent works [7], the main object of the present paper is to investigate some properties of multivalent analytic functions associated with the lemniscate of Bernoulli.

2. Main Results

The following lemma will be required.

Lemma 1 (see [8]). Let be univalent in the unit disk and let and be analytic in a domain containing with when . Set , . Suppose that

(1)either is convex or is a starlike univalent function in ,(2) for .

If is analytic in , , and satisfies

then , and is the best dominant.

Theorem 2. Let . If satisfies

then .

Proof. Let . Then is analytic in with . Condition (7) becomes that

Define the function by with . Since is the right-half of the lemniscate of Bernoulli, is a convex set and hence is a convex function. This shows that the function is starlike with respect to the origin. We now show that if the function satisfies the differential chain

For this purpose, let the functions and be defined by and . Clearly the functions and are analytic in and . Also let and be the functions defined by

It is obvious that is a starlike univalent function in and

for . By the lemma, it follows that . To complete the proof, it is left to show that

Since , it follows that

For , , clearly

A simple calculation shows that the minimum of the above expression is attained at . Hence

Thus or . This shows that and completes the proof.

Theorem 3. Let belong to the class and satisfy

Then

where , , and is the smallest root in of the equation

The result is sharp.

Proof. From (16) we can write

where is analytic and in . Differentiating both sides of (19) logarithmically, we arrive at

Put and . Then (19) implies that

With the help of the Carathéodory inequality

it follows from (21) that

where , , and

because of (21). Since is a dual function of , from (23), (24), and (25), we see that

Now let us calculate the minimum value of on the closed interval . From (26) we deduce that

where

Note that . Suppose that . Then

Hence, by virtue of the mathematical induction, we have for all and . This implies that

Further it follows from (23), (26), and (30) that

where

Note that and . If we let denote the root in of the equation , then (31) yields the desired result (17).
To see that the bound is the best possible, we consider the function

Clearly the function defined by (33) satisfies condition (16). Also it is obvious that, for ,

which shows that the bound can not be increased. The proof of the theorem is completed.

Setting , Theorem 3 reduces to the following.

Corollary 4. Let belong to the class and satisfy condition (16). Then is starlike of order    in

The result is sharp.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (no. 11571299) and the Natural Science Foundation of Jiangsu Province (no. BK20151304). The work is also supported by the Natural Science Foundation of Suqian College (no. 2015KY25).

References

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