%0 Journal Article
%T Arc Length Inequality for a Certain Class of Analytic Functions Related to Conic Regions
%A Wasim Ul-Haq
%A Muhammad Arif
%A Asfandyar Khan
%J Journal of Complex Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/407596
%X 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 [4¨C6]. 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
%U http://www.hindawi.com/journals/jca/2013/407596/