Some new applications of the fractional integral and four-parameter Mittag-Leffler function

The article reveals new applications of the four-parameter Mittag-Leffler function (MLF) in geometric function theory (GFT), using fractional calculus notions. The purpose of this study is to propose and explore a new integral operator of order λ using fractional calculus and the four-parameter MLF. The techniques of differential subordination theory are employed in order to derive certain univalence conditions for the newly defined fractional calculus operator involving the Mittag-Leffler function. In the proved theorems and corollaries of the paper, it is specified that the fractional integral operator of the four parameter MLF satisfies the conditions to be starlike and convex. It is also proved that the newly defined operator is a starlike, convex, and close-to-convex function of positive and negative orders, respectively. The geometric properties demonstrated for the fractional integral of the four-parameter MLF show that this function could be a valuable resource for developing the study of geometric functions theory, differential subordination, and superordination theory.