Consider a strictly increasing convex function $f(x)$ defined on the interval $[0,1)$ such that $f(0)=1$ and $\lim_{x\to 1^{-}}{f(x)}=+\infty$.
My question: Is the function $f(x)$ logarithmically convex (also called super-convex) in the interval $(1-\epsilon,1)$ for $\epsilon$ sufficiently small? In other words, is $\log f(x)$ a convex function in this interval?
Thanks!