On the Absolute Monotonicity of the Logarithmic of Gaussian Hypergeometric Function

It has been shown in Yang and Tian (Acta Math Sci 42B(3):847–864, 2022) that the function \(x\mapsto -\frac{d}{dx}\log {\big [(1-x)^p{{\,\mathrm{{\mathcal {K}}}\,}}(\sqrt{x})\big ]}\) is absolutely monotonic on (0, 1) if and only if \(p\ge 1/4\), where \({{\,\mathrm{{\mathcal {K}}}\,}}(r)\) is the complete elliptic integral of the first kind defined on (0, 1). This result, in this paper, will be extended to the Gaussian hypergeometric function, more precisely, the absolutely monotonic properties of \(x\mapsto \log {\big [(1-x)^s{_2F_1}(a,b;c;x)\big ]}\) will be studied. As applications, several inequalities involving the ratio of Gaussian hypergeometric function and the generalized Grötzch ring function are established.