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

This paper systematically investigates the absolute monotonicity of two function families associated with the Gaussian hypergeometric function $F(a,b;c;x)$ (where $a,b,c\in {\mathbb{R}}_{+}$): ${\mathcal{F}}_p(x)=(1-x{)}^{p}F(a,b;c;x)$ 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 $-{\mathcal{F}}_p^{\mathrm{\prime }}$, $\pm {\mathcal{G}}_p^{\mathrm{\prime }}$ and $\pm (\ln {\mathcal{F}}_p{)}^{\mathrm{\prime }}$ 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.