Abstract and Applied Analysis

Volume 2014 (2014), Article ID 565727, 6 pages

http://dx.doi.org/10.1155/2014/565727

Research Article

Differential Subordination Results for Analytic Functions in the Upper Half-Plane

Huo Tang,1,2 M. K. Aouf,³ Guan-Tie Deng,² and Shu-Hai Li¹

¹School of Mathematics and Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China

²School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

³Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 11 November 2013; Accepted 19 January 2014; Published 25 February 2014

Academic Editor: V. Ravichandran

Copyright © 2014 Huo Tang 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.

There are many articles in the literature dealing with differential subordination problems for analytic functions in the unit disk, and only a few articles deal with the above problems in the upper half-plane. In this paper, we aim to derive several differential subordination results for analytic functions in the upper half-plane by investigating certain suitable classes of admissible functions. Some useful consequences of our main results are also pointed out.

1. Introduction

Let denote the upper half-plane; that is,

and let denote the class of functions which are analytic in and which satisfy the so-called hydrodynamic normalization (see [1–3]):

Also let denote the class of all functions in which are univalent in . Various basic properties concerning functions belonging to the class were developed in a series of articles (see, for details, [4–6]).

A function , with , is said to be starlike in if and only if

We denote by the subclass of which consists of functions which are starlike in .

A function , with , is said to be convex in if and only if

Also, we denote by the subclass of which consists of functions which are convex in . The classes and were introduced by Stankiewicz [3].

We first need to recall the notion of subordination in the upper half-plane.

Let and be members of . The function is subordinate to , written as or , if there exists a function with such that . Furthermore, if the function is univalent in , then we have the following equivalence (cf. [7]):

Using methods similar to those used in the unit disk, Răducanu and Pascu [7] have extended the theory of differential subordinations to the upper half-plane. In the following, we will list some definitions and theorems, which are required to prove our main results.

Definition 1 (see [8, Definition 8.3i, p.403]). Denote by the set of functions that are analytic and injective on , where

and are such that for .

Definition 2 (see [7]). Let be a set in and . The class of admissible functions consists of those functions that satisfy the following admissibility condition:

whenever

where , , and .

If , then the admissibility condition reduces to

where , , and .

Theorem 3 (see [7]). Let and . If

for , then

In the present paper, by making use of the differential subordination results in the upper half-plane of Rǎducanu and Pascu [7] (which is a generalization of results in the unit disk obtained by Miller and Mocanu [8]), we determine certain appropriate classes of admissible functions and investigate some differential subordination properties of analytic functions in the upper half-plane. It should be remarked in passing that, in recent years, several authors obtained many interesting results associated with differential subordination and superordination in the unit disk; the interested reader may refer to, for example, [9–18].

2. The Main Subordination Results

We first define the following class of admissible functions that are required in proving our first result.

Definition 4. Let be a set in and . The class of admissible functions consists of those functions that satisfy the following admissibility condition:

whenever

Theorem 5. Let . If satisfies

then

Proof. Define the function in by

A simple calculation yields

Further computations show that

We now define the transformation from to by

Let

Using (16)–(18), and from (20), we obtain

Hence, (14) becomes

From (19), we easily get

Thus, the admissibility condition for in Definition 4 is equivalent to the admissibility condition for as given in Definition 2. Therefore , and by Theorem 3, we have , or, equivalently, , which evidently completes the proof of Theorem 5.

If is a simply connected domain, then for some conformal mapping of onto . In this case, the class is written as . The following result is an immediate consequence of Theorem 5.

Theorem 6. Let . If satisfies

then

Our next result is an extension of Theorem 5 to the case where the behavior of on is not known.

Theorem 7. Let and be univalent in with , and set and . Let satisfy one of the following conditions:

( 1), for some , or(2)there exists such that for all .

If satisfies (24), then

Proof. The proof of Theorem 7 is similar to that of [8, Theorem 2.3d, p.30] and so we choose to omit it.

The next theorem yields the best dominant of the differential subordination (24).

Theorem 8. Let be univalent in and . Suppose that the following differential equation:

has a solution and satisfies one of the following conditions:

(1) and ,(2) is univalent in and , for some , or(3) is univalent in and there exists such that for all .

If satisfies (24), then

and is the best dominant.

Proof. Following the same arguments as in [8, Theorem 2.3e, p.31], we deduce that is a dominant from Theorems 6 and 7. Since satisfies (27), it is also a solution of (24) and therefore will be dominated by all dominants. Hence, is the best dominant.

In the particular case , and in view of Definition 4, the class of admissible functions , denoted by , is described below.

Definition 9. Let be a set in . The class of admissible functions consists of those functions such that

whenever , , , and .

Corollary 10. Let . If satisfies

then

For the special case , the class is simply denoted by . Corollary 10 can now be written in the following form.

Corollary 11. Let . If satisfies

then

Example 12. Let the functions be analytic in and satisfy and . Then, the functions

satisfy the admissibility condition (29) and hence Corollary 10 yields

Next, we introduce the following class of admissible functions.

Definition 13. Let be a set in and . The class of admissible functions consists of those functions that satisfy the admissibility condition

where , , and .

Theorem 14. Let . If satisfies

then

Proof. Define the function in by

A simple calculation yields

Define the transformation from to by

Let

The proof will make use of Theorem 3. Using (39) and (40), and from (42), we get

Hence, (37) becomes

From (42), we see that the admissibility condition for in Definition 13 is equivalent to the admissibility condition for as given in Definition 2. Hence , and by Theorem 3, we have or .

We will denote the class by , where is the conformal mapping of onto .

Theorem 15. Let . If satisfies

then

We extend Theorem 15 to the case in which the behavior of on is not known.

Theorem 16. Let and be univalent in with . Let , for some , where . If satisfies (37), then (46) holds.

As a special case, when , we get the following corollary.

Corollary 17. Let be a set in and let satisfy

whenever , , and . If satisfies

then

In the special case , Corollary 17 reduces to the following corollary.

Corollary 18. Let satisfy

whenever , , and . If satisfies

then

Example 19. Let the function be analytic in and satisfy . Then, the function

satisfies the admissibility condition (47) and hence Corollary 18 becomes

Conflict of Interests

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

Acknowledgments

The present investigation was partly supported by the Natural Science Foundation of China under Grant 11271045, the Higher School Doctoral Foundation of China under Grant 20100003110004, the Natural Science Foundation of Inner Mongolia of China under Grant 2010MS0117, and the Higher School Foundation of Inner Mongolia of China under Grant NJZY13298. The authors would like to thank Professor V. Ravichandran for his valuable suggestions and the referees for their careful reading and helpful comments to improve their paper.

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