Inequalities of extended Beta and extended hypergeometric functions

TL;DR

This paper investigates the properties of extended beta and hypergeometric functions, establishing inequalities and monotonicity results that extend classical functions and provide new bounds and reverse inequalities.

Contribution

It introduces new Turán-type inequalities and monotonicity properties for extended beta and hypergeometric functions, including special cases and reverse inequalities.

Findings

Established log-convexity of extended beta functions

Derived Turán-type inequalities for extended hypergeometric functions

Proved monotonicity, log-convexity, and log-concavity properties

Abstract

This paper studies the log-convexity of the extended beta functions. As a consequence, Tur\'an-type inequalities are established.The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the inequalities on extended beta functions,. The particular cases of those results also gives the Tur\'an-type inequalities for extended confluent and extended Gaussian hypergeometric functions. Some reverse of Tur\'an-type inequalities also derived.