Journal of Applied Mathematics

Volume 2013 (2013), Article ID 869469, 4 pages

http://dx.doi.org/10.1155/2013/869469

Research Article

Some Sufficient Conditions for Starlikeness and Convexity of Order

Yi-Ling Cang¹ and Jin-Lin Liu²

¹Department of Mathematics, Suqian College, Suqian 223800, China

²Department of Mathematics, Yangzhou University, Yangzhou 225002, China

Received 22 November 2012; Revised 12 February 2013; Accepted 16 February 2013

Academic Editor: Zhijun Liu

Copyright © 2013 Yi-Ling Cang 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.

We derive certain sufficient conditions for starlikeness and convexity of order of analytic functions in the unit disk. Applications are indicated for the subordination results to electromagnetic cloaking.

1. Introduction

Let denote a class of functions of the form

which are analytic in the open unit disk . When , we write .

Let and be analytic in . Then, the function is said to be subordinate to , written as

if there exists a Schwarz function with and such that . Furthermore, if the function is univalent in , then

A function in is said to be starlike of order in if it satisfies

or equivalently

for some real (). We denote by the subclass of consisting of all starlike functions of order in .

A function is said to be convex of order in if it satisfies

or equivalently

for some real . We denote by the subclass of consisting of all functions which are convex of order in . Also, we denote that , and , respectively.

There are many results for conditions for to be in the classes and (e.g., see [1–11]). In the present paper, we aim at deriving some sufficient conditions for starlikeness and convexity of order of functions in . In particular, we extend some related results obtained by several authors [4–6, 10]. Possible applications of the subordination results to electromagnetic cloaking are also discussed in Section 3.

To derive our results, we need the following lemmas.

Lemma 1 (see [3]). Let be analytic and nonconstant in with . If and , then

Lemma 2 (see [4]). If satisfies

then .

Lemma 3 (see [4]). If satisfies

then , where is the unique root of the equation

2. Main Results

Our first result is contained in the following.

Theorem 4. If satisfies in and

for some real , then

which is equivalent to .

Proof. Let us define the analytic function in by

Then, , , and

Suppose that there exists a point such that

where is real and . Then, applying Lemma 1, we get

Thus, it follows from (15), (16), and (17) that

In view of , from (17) and (18), we obtain

But both (19) and (20) contradict assumption (12). Therefore, we must have ; that is,

The proof of the theorem is complete.

By taking in Theorem 4, we have the following result which is due to Lin and Owa [2].

Corollary 5. If satisfies in and

then .

Next, we derive the following.

Theorem 6. If satisfies

for some real , then

or equivalently .

Proof. For , we define the function by

Then, is analytic in , and

according to the condition of the theorem. By using Lemma 2, we have that .

Note that

This shows that

that is, . The proof of the theorem is complete.

Theorem 7. If satisfies

for some real , then

or equivalently , where is the unique root of the equation

Proof. Let us define the function as in (25). Then, we have

In view of Lemma 3, we see that if

then . This shows that .

Finally, we discuss the following theorem.

Theorem 8. If satisfies

for some real , then

or equivalently , where is the unique root of the equation

Proof. For , we define the function by

Then, is analytic in . Further, letting , we obtain that

Thus, applying Lemma 3, we have that

for , which shows that . This gives us that ; that is, .

3. Applications

To make objects invisible to human eyes has been long for a subject of science fiction. But just in 2006, this imagination has been materialized in the range of microwave radiation. This is attributed to pioneering papers published in Science by Leonhardt [12] and Pendry et al. [13] in 2006, in which they proposed an ingenious idea to control electromagnetic waves by specially designed materials. They suggest that a cloak, made of metamaterial (in which the refractive index spatially varies), can be designed so that an incident electromagnetic wave can be guided through the cloak giving an impression of free space when viewed from outside. This ensures that the cloak neither reflects nor scatters waves nor casts a shadow in the transmitted field. The cloak remains undetected by a viewing device. At the same time, the cloak reduces scattering of radiation from the object where the imperfections are exponentially small. Hence, the object becomes invisible to the detector.

Reports are available in the published literature (e.g., see [12, 13]) that electromagnetic cloaking, which seemed impossible earlier, is technologically realizable when the cloak and the cloaked object have a circular symmetry in at least one plane, namely, spheres and cylinders. The cross-section is a laminar or two-dimensional cloaking. For some important research contributions on this subject, see, for example, [14–17].

Mathematically, the two-dimensional cloak and the cloaked object are simply connected regions in the complex plane, the latter being a subset of the former. By the Riemann mapping theorem, both regions are equivalent to conformal maps on the unit disk . If we denote the cloaked object by the function and the cloak by the function , then it is required that

Very recently, Mishra et al. [18] have given some applications of subordination relationship (40) to electromagnetic cloaking. In the present paper, we found several sufficient conditions under which relationship of the form (40) holds for functions which are more general than circular maps. For example, in Theorem 4, we have taken the cloak function

to be an analytic univalent convex map. If a function satisfies condition (12), then the subordination relationship (40) holds true.

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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