Abstract
This article introduces a generalization for the Srivastava-Owa fractional operators in the unit disk. Conditions are given for the fractional integral operator to be bounded in Bergman space. Some properties for the above operator are also provided. Moreover, applications of these operators are posed in the geometric functions theory and fractional differential equations.
Highlights
The theory of fractional calculus has found interesting applications in the theory of analytic functions
4.2 Fractional differential equations we focus our attention on the fractional differential equation of the form
We found that the generalize integral operator satisfying the semi-group property. Their applications appeared in the theory of geometric functions and fractional differential equations by establishing the sufficient conditions for the existence and uniqueness of Cauchy problem in the unit disk
Summary
The theory of fractional calculus has found interesting applications in the theory of analytic functions. The classical definitions of fractional operators and their generalizations have fruitfully been applied in obtaining, for example, the characterization properties, coefficient estimates [1], distortion inequalities [2] and convolution structures for various subclasses of analytic functions and the works in the research monographs. In [3], Srivastava and Owa gave definitions for fractional operators (derivative and integral) in the complex z-plane C as follows: Definition 1.1. The fractional derivative of order a is defined, for a function f(z), by. The fractional integral of order a is defined, for a function f(z), by. Further properties of these operators can be found in [4,5]
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