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

Let $F(a,b; a+b;x)$ be the Gaussian hypergeometric function and $F_{p}(x)=(1-x)^{p}\exp ({F(a,b;a+b;x)})$ . 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 $-(\ln F_{p})'$ , $\ln F_{p}$ , $-(F_{p})'$ and $F_{p}$ . These results ultimately yield several new inequalities involving the zero-balanced Gaussian hypergeometric functions.