New Applications of Gaussian Hypergeometric Function for Developments on Third-Order Differential Subordinations

Abstract

The main objective of this paper is to present classical second-order differential subordination knowledge extended in this study to include new results regarding third-order differential subordinations. The focus of this study is on the main problems examined by differential subordination theory. Hence, the new results obtained here reveal techniques for identifying dominants and the best dominant of certain third-order differential subordinations. Another aspect of novelty is the new application of the Gaussian hypergeometric function. Novel third-order differential subordination results are obtained using the best dominant provided by the theorems and the operator previously defined as Gaussian hypergeometric function’s fractional integral. The research investigation is concluded by giving an example of how the results can be implemented.

New Developments on the Theory of Third-Order Differential Superordination Involving Gaussian Hypergeometric Function

“…A new approach to third-order differential subordination has been taken in recent studies considering another fundamental concept of the theory of differential subordination, which is the best dominant of the differential subordination. In [21,22], methods for determining the dominant of a third-order differential subordination are described, and new techniques for determining a third-order differential subordination's best dominant are also indicated.…”