JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 869469 10.1155/2013/869469 869469 Research Article Some Sufficient Conditions for Starlikeness and Convexity of Order α Cang Yi-Ling 1 Liu Jin-Lin 2 Liu Zhijun 1 Department of Mathematics Suqian College Suqian 223800 China sqc.edu.cn 2 Department of Mathematics Yangzhou University Yangzhou 225002 China yzu.edu.cn 2013 13 3 2013 2013 22 11 2012 12 02 2013 16 02 2013 2013 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.

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

1. Introduction

Let A(p) denote a class of functions f(z) of the form (1)f(z)=zp+n=1ap+nzp+n(pN={1,2,3,}), which are analytic in the open unit disk U={zC:|z|<1}. When p=1, we write A=A(1).

Let f(z) and g(z) be analytic in U. Then, the function f(z) is said to be subordinate to g(z), written as (2)f(z)g(z)(zU), if there exists a Schwarz function w(z) with w(0)=0 and |w(z)|<1(zU) such that f(z)=g(w(z))(zU). Furthermore, if the function g(z) is univalent in U, then (3)f(z)g(z)(zU)f(0)=g(0),f(U)g(U).

A function f(z) in A(p) is said to be starlike of order α in U if it satisfies (4)Re(zf(z)f(z))>α(zU) or equivalently (5)zf(z)f(z)p+(p-2α)z1-z(zU) for some real α (0α<p). We denote by Sp*(α) the subclass of A(p) consisting of all starlike functions of order α in U.

A function f(z)A(p) is said to be convex of order α in U if it satisfies (6)Re(1+zf′′(z)f(z))>α(zU) or equivalently (7)1+zf′′(z)f(z)p+(p-2α)z1-z(zU) for some real α(0α<p). We denote by Kp(α) the subclass of A(p) consisting of all functions which are convex of order α in U. Also, we denote that Sp*(0)=Sp*,S1*(0)=S*, and K1(0)=K, respectively.

There are many results for conditions for f(z)A to be in the classes S* and K (e.g., see ). In the present paper, we aim at deriving some sufficient conditions for starlikeness and convexity of order α of functions in A(p). In particular, we extend some related results obtained by several authors [46, 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 [<xref ref-type="bibr" rid="B6">3</xref>]).

Let p(z) be analytic and nonconstant in U with p(0)=1. If 0<|z0|<1 and Rep(z0)=min|z||z0|Rep(z), then (8)z0p(z0)-|1-p(z0)|22(1-  Rep(z0)).

Lemma 2 (see [<xref ref-type="bibr" rid="B8">4</xref>]).

If f(z)A satisfies (9)|f(z)-1|<25(zU), then f(z)S*.

Lemma 3 (see [<xref ref-type="bibr" rid="B8">4</xref>]).

If f(z)A satisfies (10)|argf(z)|<π2δ(zU), then f(z)S*, where δ=0.6165 is the unique root of the equation (11)2tan-1(1-δ)+(1-2δ)π=0.

2. Main Results

Our first result is contained in the following.

Theorem 4.

If f(z)A(p) satisfies f(z)f(z)0 in 0<|z|<1 and (12)|Im(z2f′′(z)f(z)-λzf′′(z)f(z))|<λ(2p+3λ)(zU) for some real λ(λ>0), then (13)zf(z)f(z)p1+z1-z(zU), which is equivalent to f(z)Sp*.

Proof.

Let us define the analytic function h(z) in U by (14)h(z)=zf(z)pf(z). Then, h(0)=1, h(z)0, and (15)z2f′′(z)f(z)-λzf′′(z)f(z)=zf′′(z)f(z)(zf(z)f(z)-λ)=(ph(z)-λ)(zh(z)h(z)+h(z)-1).

Suppose that there exists a point z0(0<|z0|<1) such that (16)Reh(z)>0(|z|<|z0|),h(z0)=iβ, where β is real and β0. Then, applying Lemma 1, we get (17)z0h(z0)-1+β22. Thus, it follows from (15), (16), and (17) that (18)I0=Im(z02f′′(z0)f(z0)-λz0f′′(z0)f(z0))=-β(p+λ)+λβz0h(z0). In view of λ>0, from (17) and (18), we obtain (19)I0-λ+(2p+3λ)β22βλ(2p+3λ)(β<0),(20)I0-λ+(2p+3λ)β22β-λ(2p+3λ)(β>0). But both (19) and (20) contradict assumption (12). Therefore, we must have Reh(z)>0(zU); that is, (21)zf(z)f(z)p1+z1-z(zU). The proof of the theorem is complete.

By taking p=λ=1 in Theorem 4, we have the following result which is due to Lin and Owa .

Corollary 5.

If f(z)A satisfies f(z)f(z)0 in 0<|z|<1 and (22)|Im{z2f′′(z)f(z)-zf′′(z)f(z)}|<5(zU), then f(z)S*.

Next, we derive the following.

Theorem 6.

If f(z)A(p) satisfies (23)|(f(z)zp)1/(p-α)(zf(z)f(z)-α)-p+α|<25(p-α)(zU) for some real α(0α<p), then (24)zf(z)f(z)p+(p-2α)z1-z(zU), or equivalently f(z)Sp*(α).

Proof.

For f(z)A(p), we define the function p(z) by (25)p(z)=(f(z)zα)1/(p-α)=z+. Then, p(z) is analytic in U, and (26)|p(z)-1|=1p-α|(f(z)zp)1/(p-α)(zf(z)f(z)-α)-p+α|<25(zU) according to the condition of the theorem. By using Lemma 2, we have that p(z)S*.

Note that (27)zp(z)p(z)=1p-α(zf(z)f(z)-α). This shows that (28)zf(z)f(z)p+(p-2α)z1-z(zU); that is, f(z)Sp*(α). The proof of the theorem is complete.

Theorem 7.

If f(z)A(p) satisfies (29)|arg(f(z)zp)+(p-α)arg(zf(z)f(z)-α)|<π2(p-α)δ(zU) for some real α(0α<p), then (30)zf(z)f(z)p+(p-2α)z1-z(zU), or equivalently f(z)Sp*(α), where δ=0.6165 is the unique root of the equation (31)2tan-1(1-δ)+(1-2δ)π=0.

Proof.

Let us define the function p(z) as in (25). Then, we have (32)argp(z)=arg(1p-α(f(z)zp)1/(p-α)(zf(z)f(z)-α))=1p-αarg(f(z)zp)+arg(zf(z)f(z)-α). In view of Lemma 3, we see that if (33)1p-α|arg(f(z)zp)+(p-α)arg(zf(z)f(z)-α)|<π2δ(zU), then p(z)S*. This shows that f(z)Sp*(α).

Finally, we discuss the following theorem.

Theorem 8.

If f(z)A(p) satisfies (34)|arg(f(z)zp-1)+(p-α)arg(1+zf′′(z)f(z)-α)|<π2(p-α)δ(zU) for some real α(0α<p), then (35)1+zf′′(z)f(z)p+(p-2α)z1-z(zU), or equivalently f(z)Kp(α), where δ=0.6165 is the unique root of the equation (36)2tan-1(1-δ)+(1-2δ)π=0.

Proof.

For f(z)A(p), we define the function p(z) by (37)p(z)=0z(f(t)ptp-1)1/(p-α)dt=z+. Then, p(z) is analytic in U. Further, letting g(z)=zp(z), we obtain that (38)g(z)=1p-α(f(z)pzp-1)1/(p-α)(1+zf′′(z)f(z)-α).

Thus, applying Lemma 3, we have that (39)|arg(g(z))|=|1p-αarg(f(z)zp-1)+arg(1+zf′′(z)f(z)-α)|<π2δ for zU, which shows that g(z)S*. This gives us that p(z)K; that is, f(z)Kp(α).

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  and Pendry et al.  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, .

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 U. If we denote the cloaked object by the function g(z) and the cloak by the function q(z), then it is required that (40)g(z)q(z)(zU).

Very recently, Mishra et al.  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 (41)q(z)=p1+z1-z to be an analytic univalent convex map. If a function f(z) 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.

Li J.-L. Owa S. Sufficient conditions for starlikeness Indian Journal of Pure and Applied Mathematics 2002 33 3 313 318 MR1894627 ZBL0998.30010 Lin L. J. Owa S. Properties of the Salagean operator Georgian Mathematical Journal 1998 5 4 361 366 10.1023/A:1022159619351 MR1639004 ZBL0924.30008 Miller S. S. Mocanu P. T. Second-order differential inequalities in the complex plane Journal of Mathematical Analysis and Applications 1978 65 2 289 305 10.1016/0022-247X(78)90181-6 MR506307 ZBL0367.34005 Mocanu P. T. Some starlikeness conditions for analytic functions Revue Roumaine de Mathématiques Pures et Appliquées 1988 33 1-2 117 124 MR948445 ZBL0652.30004 Mocanu P. T. Some simple criteria for starlikeness and convexity Libertas Mathematica 1993 13 27 40 MR1241346 ZBL0793.30008 Nunokawa M. Owa S. Polatoglu Y. Caglar M. Duman E. Y. Some sufficient conditions for starlikeness and convexity Turkish Journal of Mathematics 2010 34 3 333 337 MR2681578 ZBL1195.30030 Ramesha C. Kumar S. Padmanabhan K. S. A sufficient condition for starlikeness Chinese Journal of Mathematics 1995 23 2 167 171 MR1340763 ZBL0834.30010 Sokół J. On an application of certain sufficient condition for starlikeness Journal of Mathematics and Applications 2008 30 131 135 MR2446478 ZBL1155.30009 Sokól J. On a condition for α-starlikeness Journal of Mathematical Analysis and Applications 2009 352 2 696 701 10.1016/j.jmaa.2008.11.025 MR2501913 Uyanık N. Aydogan M. Owa S. Extensions of sufficient conditions for starlikeness and convexity of order α Applied Mathematics Letters 2011 24 8 1393 1399 10.1016/j.aml.2011.03.018 MR2793640 ZBL1216.30023 Xu N. Yang D. Some criteria for starlikeness and strongly starlikeness Bulletin of the Korean Mathematical Society 2005 42 3 579 590 10.4134/BKMS.2005.42.3.579 MR2162214 ZBL1094.30021 Leonhardt U. Optical conformal mapping Science 2006 312 5781 1777 1780 10.1126/science.1126493 MR2237569 ZBL1226.78001 Pendry J. B. Schurig D. Smith D. R. Controlling electromagnetic fields Science 2006 312 5781 1780 1782 10.1126/science.1125907 MR2237570 ZBL1226.78003 Kundtz N. Smith D. R. Extreme-angle broadband metamaterial lens Nature Materials 2010 9 129 132 Liu R. Ji C. Mock J. J. Chin J. Y. Cui T. J. Smith D. R. Broadband ground-plane cloak Science 2009 323 5912 366 369 10.1126/science.1166949 Rao G. A. Mahulikar S. P. Integrated review of stealth technology and its role in airpower Aeronautical Journal 2002 106 629 641 Schurig D. Mock J. J. Justice B. J. Cummer S. A. Pendry J. B. Starr A. F. Smith D. R. Metamaterial electromagnetic cloak at microwave frequencies Science 2006 314 5801 977 980 Mishra A. K. Panigrahi T. Mishra R. K. Subordination and inclusion theorems for subclasses of meromorphic functions with applications to electromagnetic cloaking Mathematical and Computer Modelling 2013 57 945 962