Convexity properties related to Gauss hypergeometric function

Abstract

We investigate the convexity property on $(0,1)$ of the functions ${\phi }_{a,b,c}$ and $1/{\phi }_{a,b,c}$, where $${\phi }_{a,b,c}(x)=\frac{c-\log (1-x)}{{}_2{F}_1(a,b,a+b,x)},$$ whenever $a,b\ge 0$ and $a+b\le 1$. We Show that ${\phi }_{a,b,c}$ (respectively $1/{\phi }_{a,b,c}$) is strictly convex on $(0,1)$ if and only if $c\le -2\gamma -\psi (a)-\psi (b),$ (respectively $c\ge {\alpha }_0$) and ${\phi }_{a,b,c}$ (respectively $1/{\phi }_{a,b,c}$) is strictly concave on $(0,1)$ if and only if $c\ge c(a,b)$ (respectively $c\in [{\delta }_{-},{\delta }_{+}]$), where $\psi$ is the Polygamma function. This generalizes some problems posed by Yang and Tian and complete the study of convexity properties of functions studied by the author in [bouali]. As applications of the convexity and concavity, we establish among other inequalities, that for all $x\in (0,1)$, $a,b\in [0,1]$, $a+b\le 1$ and $c\ge c(a,b)$ $$c+\frac{\Gamma (a)\Gamma (b)}{\Gamma (a+b)}\le \frac{c-\log (1-x)}{{}_2{F}_1(a,b,a+b,x)}+\frac{c-\log (x)}{{}_2{F}_1(a,b,a+b,1-x)}\le \frac{(2c+2\log 2)}{{}_2{F}_1(a,b;a+b;1/2)},$$ and for all $x\in (0,1)$, $a,b\in [0,1]$, $a+b\le 1$ and $c\in [{\delta }_{-},{\delta }_{+}]$ $$\frac{1}{c}+\frac{\Gamma (a+b)}{\Gamma (a)\Gamma (b)}\le \frac{{}_2{F}_1(a,b,a+b,x)}{c-\log (1-x)}+\frac{{}_2{F}_1(a,b,a+b,1-x)}{c-\log (x)}\le \frac{{}_2{F}_1(a,b;a+b;1/2)}{(2c+2\log 2)}.$$