Absolutely monotonic functions involving the zero-balanced Gaussian hypergeometric functions with applications

Abstract

Abstract Let F ( a , b ; a + b ; x ) $F(a,b; a+b;x)$ be the Gaussian hypergeometric function and F p ( x ) = ( 1 − x ) p exp ( F ( a , b ; a + b ; x ) ) $F_{p}(x)=(1-x)^{p}\exp ({F(a,b;a+b;x)})$ . This article aims to extend the work of Zhen-Hang Yang and Jing-Feng Tian to a more generalized case involving zero-balanced Gaussian hypergeometric functions. We prove the sufficient and necessary conditions for the absolute monotonicity of − ( ln F p ) ′ $-(\ln F_{p})'$ , ln F p $\ln F_{p}$ , − ( F p ) ′ $-(F_{p})'$ and F p $F_{p}$ . These results ultimately yield several new inequalities involving the zero-balanced Gaussian hypergeometric functions.

Keywords

• Absolute monotonicity
• Zero-balanced
• Gaussian hypergeometric function
• Inequality