Necessary and Sufficient Conditions for Absolute Monotonicity of Functions Related to Gaussian Hypergeometric Functions

Abstract: This paper systematically investigates the absolute monotonicity of two function families associated with the Gaussian hypergeometric function $F(a, b; c; x)$ (where $a,b,c\in\mathbb{R}_+$): $\mathcal{F}_p(x)=(1-x)^pF(a,b;c;x)$ and $\mathcal{G}_p(x)=(1-x)^p \exp(F(a,b;c;x))$, as well as the logarithmic transform $\ln\mathcal{F}_p(x)$. Our primary goal is to establish necessary and sufficient conditions for the parameter $p$ such that $-\mathcal{F}'_p$, $\pm\mathcal{G}'_p$ and $\pm(\ln\mathcal{F}_p)'$ are absolutely monotonic on $(0,1)$. Additionally, we derive several results regarding the absolute monotonicity of their higher-order derivatives. As applications, we derive several new inequalities for the Gaussian hypergeometric function $F(a,b;c;x)$. Most importantly, we develop a novel constructive approach based on Jurkat's criterion for power series ratios, which avoids limitations of cumbersome recursive/inductive methods in existing literature.