Table of Contents
Journal of Complex Analysis
Volume 2013, Article ID 407596, 4 pages
http://dx.doi.org/10.1155/2013/407596
Research Article

Arc Length Inequality for a Certain Class of Analytic Functions Related to Conic Regions

Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan

Received 22 May 2013; Accepted 26 July 2013

Academic Editor: Narendra Govil

Copyright © 2013 Wasim Ul-Haq 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

In our present investigation, we introduce a subclass of analytic function associated with conic regions which is a form of generalized close-to-convexity. The arc-length inequality for a class of analytic function is well known. We derive this inequality for the newly defined class and also study some of its interesting consequences.

1. Introduction

Let denote the class of functions : which are analytic in the unit disc . Let denote the class of all functions in which are univalent. Also let , , and be the well-known subclasses of consisting of all functions which are, respectively, of starlike, convex, and close-to-convex.

Kanas and Wisniowska [1, 2] studied the classes of -uniformly convex denoted by and the corresponding class related by the Alexandar type relation. Later Acu [3] considered the class -uniformly close-to-convex denoted by to be defined as for more detail see [46].

In [7], the conic domain with complex order is defined as where The domain is elliptic for , hyperbolic when , parabolic for , and right half plane when . The functions which play the role of extremal functions for the conic regions of complex order are given as where ,  , , and is chosen such that , where is the Legendre's complete elliptic integral of the first kind and is complementary integral of , see [1, 2].

Let be the class of functions with positive real part, and let be the subclass of containing the functions , such that . Motivated from Noor’s work [8], we extend class to class ,   which is defined as

Note that and , the class introduced and studied by Pinchuk [9].

We define the following class: where Geometrically, a function means that the functional takes all the values in conic domain and its boundary rotation is at most . We note that class coincides with already known classes of analytic functions by choosing special values for the involved parameters. For example, for , we have the class introduced and studied by Noor [10], and further along with this by taking , we obtain the well-known class of close-to-convex functions. The purpose of this paper is to investigate some interesting properties of class . For this, we require the following results.

Lemma 1. A function if and only if(i), , (ii)there exist two normalized starlike functions and such that

The above lemma can be proved by using the similar procedure as in [11]; also see [8].

Lemma 2 (see [12]). Let with . Then,

2. Some Properties of the Class

In this section, we provide some of the interesting properties of class such as radius of convexity problem, arc length, and growth rate of its coefficients. The following theorem is readily seen when we proceed on similar lines as in [13].

Theorem 3. The function if and only if where and are close-to-convex functions.

Theorem 4. Let in . Then, for , where This result is sharp.

Proof. We can write
Using Lemma 1(ii), we get where and are starlike functions. Logarithmic differentiation of (14) gives us
This implies that Now using distortion results for the class , we have The right hand side of (17) is positive for , where is given by (12). The sharpness can be viewed from the function , given by
We note the following interesting special cases:(i)For , we have the radius of convexity for class .(ii)For   and  , we have the radius of convexity for class , proved by Noor [10].(iii)For ,   and  , we have radius of convexity for close-to-convex functions which is well known.

Theorem 5. Let with , , and . Then, The exponent is sharp.

Proof. Let . Then, there exists such that From the definition of , one can deduce that implies that .
Now using (20), Lemma 1(ii), and distortion theorems for starlike functions, we have By using Hölder's inequality, this gives Since , therefore subordination for starlike functions and Lemma 2 give us The function is defined by where shows that the exponent is sharp.

Some special choices in the above theorem give us the following interesting results.

Corollary 6. Let . Then

Corollary 7. Let . Then

Coefficient Growth Problems. The problem of growth rate and asymptotic behavior of coefficients is well known. In the next results, we study these problems for class by varying different parameters.

Theorem 8. Let with , and . Then The exponent is sharp.

Proof. With , Cauchy’s theorem gives us Using Theorem 5 and putting , we obtain the required result. The sharpness follows from the function defined by the relation (24).

Corollary 9. Let , and let it be of the form (1). Then, for ,  , one has For , in the above corollary, we have growth rate of coefficients problem for functions in class , and, for , gives us the growth rate of coefficients for close-to-convex functions, which are well known.

Acknowledgments

The authors want to acknowledge worthy referees of this paper for their insightful comments which greatly improves the entire presentation of the paper. They would also like to thank Prof. Dr. Ehsan Ali, VC AWKUM, for providing research facilities.

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