Let f be a positive function on (0, 1) such that as . Must there exist a convex function g such that with as ? Prove or give a counter example.
My Progress: Truthfully, I haven't made all that much. My intuition is that g exists. We can make a bunch of assumptions WLOG about f, e.g. that it is monotone; we can also assume that f is a step function or piecewise linear, but so far I haven't been very successful at coming up with anything even in these cases.