GENERALIZED CLASS OF ANALYTIC FUNCTIONS USING ℘-GAMMA FRACTIONAL OPERATORS OF A COMPLEX VARIABLE

The extended gamma function as an active function expands the notion of the standard gamma function to include two or more elements. The extended gamma function, on the other hand, is a mathematical function that extends the normal gamma function concept in order to incorporate two factors (2D-gamma function). This function appears in all classic fractional and fractal-fractional operators, which serves as one of its most significant applications. Based on 2D-gamma function, the aim of this work is to introduce a generalization of the Riemann–Liouville fractal-fractional operators (differential and integral), extend these operators into a complex domain (the open unit disk) to obtain complex fractal-fractional operators, normalize the complex fractal-fractional operators in order to study them geometrically in the open unit disk, consider the complex differential fractal-fractional operators in a class of fractal-fractional differential equations of the formula [Formula: see text], where [Formula: see text] is normalized analytic function [Formula: see text] and investigate the boundedness of the proposed class in the open unit disk. The findings obtained from the analyses demonstrate that the proposed class has a maximum bound by the generalized Fox–Wright function whose adaptability has been revealed, beside elucidating the geometric function theory.