This paper 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 main 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). Moreover, 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). We develop a novel constructive approach based on Jurkat’s criterion for power series ratios, which avoids limitations of cumbersome recursive/inductive methods in the existing literature.
