Journal of Function Spaces

Volume 2016 (2016), Article ID 6086409, 9 pages

http://dx.doi.org/10.1155/2016/6086409

## Fractional Calculus of Analytic Functions Concerned with Möbius Transformations

^{1}Department of Mathematics-Informatics, Faculty of Sciences, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania^{2}Department of Mathematics, Faculty of Education, Yamato University, Katayama 2-5-1, Suita, Osaka 564-0082, Japan

Received 17 October 2015; Accepted 3 March 2016

Academic Editor: Simone Secchi

Copyright © 2016 Nicoleta Breaz 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

Let be the class of functions in the open unit disk with and . Also, let be a Möbius transformation in for some . Applying the Möbius transformations, we consider some properties of fractional calculus (fractional derivatives and fractional integrals) of . Also, some interesting examples for fractional calculus are given.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disk .

If satisfiesfor some real , then is said to be starlike of order in . We denote by the class of all starlike functions of order in and . Furthermore, if satisfies for some real , then we say that is convex of order in . We also denote by the class of all such functions and . In view of definitions for the classes and , we know that(i) if and only if ;(ii) if and only if .

Further, MacGregor [1] and Wilken and Feng [2] have the sharp inclusion relation that for each with For , Marx [3] and Strohhäcker [4] showed that . Also, by Robertson [5], we know that the extremal function for the class is and the extremal function for the class is

For , we apply the following Möbius transformation:for a fixed . This Möbius transformation maps onto itself and to .

#### 2. Fractional Calculus

From among the various definitions for fractional calculus (i.e., fractional derivatives and fractional integrals) given in the literature, we have to recall here the following definitions for fractional calculus which are used by Owa [6, 7] and by Owa and Srivastava [8].

*Definition 1. *The fractional integral of order is defined, for , by where and the multiplicity of is removed by requiring to be real when .

*Definition 2. *The fractional derivative of order is defined, for , bywhere and the multiplicity of is removed as in Definition 1 above.

*Definition 3. *Under the hypotheses of Definition 2, the fractional derivative of order is defined by where and .

*Remark 4. *In view of definitions for the fractional calculus of , we see that for and .

Therefore, we can write thatfor any real number .

Using the fractional calculus (13), we defineIf we take in (14), thenimplies the Libera integral operator defined by Libera [9]. Therefore, given by (14) is the generalization operator of Libera integral operator.

Let us give two examples for the fractional operator defined in (14).

*Example 5. *Let us define byThen, we have thatwhere . If we definethenThis shows us thatThat is, Therefore, .

For , given by (16), becomesThen, we see that .

Next, let us consider the function given byfor a fixed , where is given by (7). Then, it is easy to see that . Taking in (23), we have thatLettingwe obtain thatThis shows that , , and . Therefore, there exists some such that for . It follows thatThus, we say that Consequently, we say that , for given by (16).

If , thenThe open unit disk is mapped on the starlike domain of order in Figure 1.

If , thenThus, maps onto the starlike domain of order in Figure 2.