Using the series representation of the hypergeometric function, we see that $_2F_1(a,b\,;c\,;(1+x)^{-1})$, $x>0$, is the convergent limit of positive sums of completely monotonic functions for $a,b,c>0$, and is thus completely monotonic.
Since completely monotonic functions are log convex, it follows that $\log \big({}_2F_1(a,b\,;c\,;(x+1)^{-1}\big)$ is convex on $(0,\infty)$.