Abstract and Applied Analysis

Volume 2016 (2016), Article ID 6180140, 8 pages

http://dx.doi.org/10.1155/2016/6180140

## Certain Properties of Some Families of Generalized Starlike Functions with respect to -Calculus

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 16 July 2016; Accepted 1 September 2016

Academic Editor: Jozef Banas

Copyright © 2016 Ben Wongsaijai and Nattakorn Sukantamala. 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

By making use of the concept of -calculus, various types of generalized starlike functions of order were introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types of -starlike functions of order . We also introduce and study a new subclass of -starlike functions of order . Moreover, we give some properties of those -starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class.

#### 1. Introduction and Preliminaries

The quantum calculus, so called -calculus and -calculus, is the usual calculus without using the notion of limits. The letter apparently stands for Planck’s constant and the letter obviously stands for quantum. Here, quantum calculus is not the same as quantum physics. Due to the applications in various fields of mathematics and physics, the study of -calculus has been very attractive for many researchers. Jackson [1, 2] was the first person in developing a -derivative, also a -integral, in a systematic mean. Afterward on quantum groups, the geometrical interpretation of -analysis has been studied. The relation between -analysis and integrable systems has been recognized. Based on -analogue of beta function, Aral and Gupta [3–5] defined and studied the -analogue of Baskakov Durrmeyer operator. Also, there are some discussions on -Picard and -Gauss-Weierstrass singular integral operators which are the other important -generalization of complex operators (see [6–8]).

In geometric function theory, there are many applications of -calculus on subclasses of analytic functions, especially subclasses of univalent functions. In [9], Ismail et al. first introduced the class of generalized functions via -calculus. In [10], Raghavendar and Swaminathan have studied some basic properties of -close-to-convex functions. In [11], Mohammed and Darus studied geometric properties and approximations of these -operators in some subclasses of analytic functions in the disk. By using the convolution of normalized analytic functions and -hypergeometric functions, these -operators have been defined. The inclusive study on applications of -calculus in operator theory could be seen in [12]. Recently, Esra Özkan Uçar [13] studied the coefficient inequality for -closed-to-convex functions with respect to Janowski starlike functions. Here, many newsworthy results related to -calculus and subclasses of analytic functions theory are studied by various authors (see [14–21]).

Let be the open disk radius centered at origin and the open unit disk is then defined by . We denote by the class of functions in the formwhich is analytic in and satisfying the usual normalization condition . We denote by the subclass of consisting of functions, which are univalent on . A function is said to be starlike of order in if satisfiesWe denote this class by . In particular, we set for a class of starlike functions on . Class is closely related to class ). A function is said to belong to class if satisfies

For the convenience, we provide some basic definitions and concept details of -calculus which are used in this paper. For any fixed complex number , a set is called a -geometric set if for , . Let be a function defined on a -geometric set. Jackson’s -derivative and -integral of a function on a subset of are, respectively, given by (see Gasper and Rahman [22], pp. 19–22)In case , the -derivative and -integral of , where is a positive integer, are given by As and , we have .

To generalize the class of starlike functions, it seems that replacing the derivative function , which appears in (2), by the -difference operator is an easily way to generalize the class of starlike functions. The definition turned out to be the following.

*Definition 1. *A function is said to belong to class , , ifTo put it in words, we call the class of -starlike functions of order type .

Now we recall another way to generalize the class of starlike functions proposed by Ismail et al. [9]. In their works, the usual derivative was replaced by the -difference operator . Moreover, the right-half plane was substituted by an appropriate domain. Later, Agrawal and Sahoo in [14] extended the ideas in [9] to -starlike function of order . Then the definition turned out to be the following.

*Definition 2. *A function is said to belong to class , , ifTo put it in words, we call the class of -starlike functions of order type .

In addition, we now introduce new type of -starlike functions.

*Definition 3. *A function is said to belong to class , , ifTo put it in words, we call the class of -starlike functions of order type .

The main objective of this paper is to characterize in 4 sections. In Section 2, we give some relations between such classes and a sufficient condition via coefficient inequality. In Section 3, we study some properties of those -starlike functions of order with negative coefficient. Here, some results on the radius of univalent and starlikeness order on the class of -starlike functions with negative coefficient are obtained. Some illustrative examples of radius of univalent and starlikeness on some functions with negative coefficient are demonstrated in Section 4.

#### 2. Main Results

We first show the inclusion theorem via geometric properties of each type of -starlike functions.

Theorem 4. *For , then *

*Proof. *Assuming that , by using triangle inequality and (8), we haveThen ; that is, . Next, we let . Since that is, lies in the circle of radius with a center at , and we observe that which means that , then ; that is, . This completes the proof.

Geometrically, for , , lied in the difference domains:respectively; see Figure 1.