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

Abstract

In this paper, for a suitable region of (a, b), we establish a necessary and sufficient condition of \(p>0\) such that

$$\begin{aligned} x\mapsto \frac{\log (p/\sqrt{1-x})}{F(a,b;a+b;x)} \end{aligned}$$

is strictly monotonic, convex, or concave on (0, 1), where \(F(a,b;a+b;x)\) represents the zero-balanced hypergeometric function. This extends the recently obtained corresponding results for the cases that \(a=b=1/2\). As applications, several functional inequalities involving zero-balanced hypergeometric function will be obtained.