Analysis of a Normalized Structure of a Complex Fractal–Fractional Integral Transform Using Special Functions

Abstract

By using the most generalized gamma function (parametric gamma function, or p-gamma function), we present the most generalized Rabotnov function, called the p-Rabotnov function. Consequently, new fractal–fractional differential and integral operators of a complex variable in an open unit disk are defined and investigated analytically and geometrically. We address some inequalities involving the generalized fractal–fractional integral operator in some spaces of analytic functions. A novel complex fractal–fractional integral transform (CFFIT) is presented. A normalization of the proposed CFFIT is observed in the open unit disk. Examples are illustrated for power series of analytic functions.

Keywords

• fractional calculus
• fractal calculus
• fractional difference operator
• fractal–fractional differential operator
• fractal–fractional calculus
• complex transform