Boundary fractional differential equation in a complex domain
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Boundary fractional differential equation in a complex domain Rabha W Ibrahim1* and Jay M Jahangiri2 * Correspondence: [email protected] 1 Institute of Mathematical Sciences, University Malaya, Kuala Lumpur, 50603, Malaysia Full list of author information is available at the end of the article
Abstract We discuss univalent solutions of boundary fractional differential equations in a complex domain. The fractional operators are taken in the sense of the Srivastava-Owa calculus in the unit disk. The existence of subsolutions and supersolutions (maximal and minimal) is established. The existence of a unique univalent solution is imposed. Applications are constructed by making use of a transformation formula for fractional derivatives as well as generalized fractional derivatives.
1 Introduction Fractional calculus is the most significant branch of mathematical analysis that transacts with the potential of covering real number powers or complex number powers of the differentiation operator D = d/dx. This concept was harnessed in geometric function theory (GFT). It was applied to derive different types of differential and integral operators mapping the class of univalent functions and its subclasses into themselves. Hohlov [, ] imposed sufficient conditions that guaranteed such mappings for the operators defined by means of the Hadamard product (or convolution) with Gauss hypergeometric functions. This was further extended by Kiryakova and Saigo [] and Kiryakova [, ] to the operators of the generalized fractional calculus (GFC) consisting of product functions of the Gaussian function, generalized hypergeometric functions, G-functions, Wright functions and Fox-Wright generalized functions as well as rendering integral representations by means of Fox H-functions and the Meijer G-function. These techniques can be used to display sufficient conditions that guarantee mapping of univalent functions (or, respectively, of convex functions) into univalent functions. For example, for the case of Dziok-Srivastava operator see [], and for an extension to the Wright functions see [] which is concerned with the Srivastava-Wright operator. With the help of operators introduced in [] and [] one can establish univalence criteria for a large number of operators in GFT and GFC and for many of their special cases such as operators of the classical fractional calculus. Srivastava and Owa [] generalized the definitions of fractional operators as follows. Definition . For the function f (z) analytic in a simply-connected region of the complex z-plane C containing the origin and for ≤ α < , the fractional derivative of order α is defined by Dαz f (z) :=
d ( – α) dz
z
f (ζ ) dζ , (z – ζ )α
©2014Ibrahim and Jahangiri; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly ci
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