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 \(p>0\) $p>0$ such that \(\begin{aligned} x\mapsto \frac{\log (p/\sqrt{1-x})}{F(a,b;a+b;x)} \end{aligned}\) $$\begin{array}{c}\hfill x\mapsto \frac{log(p/\sqrt{1-x})}{F(a,b;a+b;x)}\end{array}$$ is strictly monotonic, convex, or concave on (0, 1), where \(F(a,b;a+b;x)\) $F(a,b;a+b;x)$ represents the zero-balanced hypergeometric function. This extends the recently obtained corresponding results for the cases that \(a=b=1/2\) $a=b=1/2$ . As applications, several functional inequalities involving zero-balanced hypergeometric function will be obtained.