Monotonicity and convexity (concavity) properties for zero-balanced hypergeometric function

In this paper, for a suitable region of (a, b), we establish a necessary and sufficient condition of p>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0$$\end{document} such that x↦log(p/1-x)F(a,b;a+b;x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} x\mapsto \frac{\log (p/\sqrt{1-x})}{F(a,b;a+b;x)} \end{aligned}$$\end{document}is strictly monotonic, convex, or concave on (0, 1), where F(a,b;a+b;x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(a,b;a+b;x)$$\end{document} represents the zero-balanced hypergeometric function. This extends the recently obtained corresponding results for the cases that a=b=1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=b=1/2$$\end{document}. As applications, several functional inequalities involving zero-balanced hypergeometric function will be obtained.