A Modified Generalization of Fractional Calculus Operators in A Complex Domain

This investigation deals with a new generalization for fractional calculus operators in a complex domain based on the well-known hypergeometric function. Conditions are forced for these generalized operators such as the upper bounds. Other properties for the above operators are also presented. Besides, the employment of these operators is proposed in the geometric function theory.


Introduction
Fractional Calculus is a powerful tool that has been recently applied to complex mathematical with linear operators. Despite its complicated mathematical background, fractional calculus came to open a new window of opportunity to mathematical and real-world, which has appeared many new problems and acceptable results. For instance, the concepts of fractional calculus operators and their generalizations of analytic and univalent functions have been successfully obtained to determine the basic geometric properties such as the coefficient estimates and distortion inequalities for numerous subclasses of analytic functions, adding to that studied some their topological properties in a complex plane (see [1][2][3]).
In [4] introduced an approach of the fractional integral operator defined for | | and real numbers , by where the function is analytic in a simply-connected region of the − plane containing the origin, with the order | | for { } and the multiplicty of is removed by requiring to be real when . Here, is the Gamma function and is the absolutely convergent Gauss hypergeometric function given for by the power series [5]: Recently, [6] defined a modification of the fractional integral and differential operators of order two parameters and such that respectively, are presented as follows: ∫ and ∫ where the function is analytic in a simply-connected region of the − plane containing the origin, both of the multiplicity of and are respectively removed by requiring to be real when .
In this study, we shall restrict our attention to define new fractional calculus operators in the complex plane. The upper bounds for these operators given in terms of the univalent and convex functions. Some geometric applications associated with the Bessel function of the first kind are presented by the generalized Wright functions in the sense of generalization.

New classes of generalized fractional calculus operators
In this section, we proposed to define generalized fractional integral and differential operators in the classical definitions, where the order of the fractional integral and fractional differential operators must be positive real numbers. Our definition has been based on important remarks concerning in equations (2) and (3). Now, we employ equation (1) in (2) to introduce a new generalized fractional integral operator as follows:

Definition 1. Let and be real numbers and such that
Then the fractional integral operator is defined by where the function is analytic in a simply-connected region of the − plane containing the origin with the order | | for { } and the multiplicity of is removed as in equations (2) .  (1) in (3) to define a new generalized fractional differential operator by the following formula.

Definition 2.
Let and be real numbers and such that . The generalized fractional differential operator is defined by: where the function is analytic in a simply-connected region of the − plane containing the origin with order as given by (3).

Remark 2.
By setting in (5), then we obtain the following closed results: .
We shall need the following Definition to present the next outcomes in our investigation. Also, we use the familiar Gauss equation The next results are based on two formulas of generalized fractional integral (4) and generalized fractional differential (5) with a power function. Similarly to the proof of Lemma 1, it is proved the association of the generalized fractional differential operator (5)

Upper Bounds
In this section, we deal with some applications of the new generalizations of fractional operators (4) and (5) (8)  The following Theorem 6 introduces the generalized fractional differential operator (5) of the Bessel function (9). Theorem 6. Let be positive non-zero numbers, and be such that . Then

Conclusion
Conditions for the new fractional calculus operators are obtained. Also, some characteristics for these operators are delivered. Some geometric applications are studied in the sense of generalization.