This problem is actually the Exercise 8 in Chapter 3 of Rudin's Real and Complex analysis book. The problem is as follows:
If $g$ is a positive function on $(0,1)$ such that $g(x) \to \infty$ as $x \to 0$. Does there exist a convex function $h$ on $((0,1)$ such that $h(x) \leq g(x)$ for all $x \in (0,1)$ and $h(x) \to \infty$ as $x \to 0$?
My guess is 'Yes', there does exist such a function but can not prove it. Any help will be appreciated. Thanks in advance!