Comprehensive notes on various effects of some operators of fractional-order derivatives to certain functions in the complex domains and some of related implications

Abstract

The main objective of this comprehensive research note is to reacquaint certain necessary information with respect to various type operators designated by the fractional-order calculus in certain domains of the complex plane, then to reveal various significant effects specified by those fractional-order (type) operators for certain complex functions regular in the open unit disk, and also to center upon numerous possible implications of our main results, and to put emphasis on a number of special results of related implications in relation with several differential type analytic-geometric properties of those regular functions in the concerned open set.