Convexity properties related to Gauss hypergeometric function

Abstract: We investigate the convexity property on $(0,1)$ of the functions $\varphi_{a,b,c}$ and $1/\varphi_{a,b,c}$, where $$\varphi_{a,b,c}(x)= \frac{c-\log(1-x)}{\,2F_1(a,b,a+b,x)},$$ whenever $a,b\geq 0$ and $a+b\leq 1$. We Show that $\varphi{a,b,c}$ (respectively $1/\varphi_{a,b,c}$) is strictly convex on $(0,1)$ if and only if $c\leq -2\gamma-\psi(a)-\psi(b),$ (respectively $c\geq\alpha_0$) and $\varphi_{a,b,c}$ (respectively $1/\varphi_{a,b,c}$) is strictly concave on $(0,1)$ if and only if $c\geq c(a,b)$ (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 $x\in(0,1)$, $a,b\in[0,1]$, $a+b\leq 1$ and $c\geq c(a,b)$ $$c+\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\leq \frac{c-\log(1-x)}{\,2F_1(a,b,a+b,x)}+\frac{c-\log(x)}{\,_2F_1(a,b,a+b,1-x)}\leq\frac{(2c+2\log 2)}{\,_2{F}_1(a,b;a+b;1/2)},$$ and for all $x\in(0,1)$, $a,b\in[0,1]$, $a+b\leq 1$ and $c\in [\delta-,\delta_+]$ $$\frac1c+\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\leq \frac{\,_2F_1(a,b,a+b,x)}{c-\log(1-x)}+\frac{\,_2F_1(a,b,a+b,1-x)}{c-\log(x)}\leq\frac{\,_2{F}_1(a,b;a+b;1/2)}{(2c+2\log 2)}.$$