$K^{th}$-order Differential Subordination Results of Analytic Functions in the Complex Plane

Abstract

In recent years, there have been many interesting usages for differential subordinations of analytic functions in Geometric Function Theory of Complex Analysis. The concept of the first and second-order differential subordination have been pioneered by Miller and Mocanu. In 2011, the third-order differential subordination were defined to give a new generalization to the concept of differential subordination. While the fourth-order differential subordination has been introduced in 2020. In the present article, we introduce new concept that is the Kth-order differential subordination of analytic functions in the open unit disk U.