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.