If the problem is completely stated, examine the function
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.
Whoops, the question isn't stated fully. We require as as well. So, we need f to be bounded from below by convex g where g also goes to infinity. As originally stated, obviously would work.
I don't think either of the prior posts give any help; obviously not the posters fault since the question was incomplete.
Yes, sorry, the modified wording was still wrong. The functions are positive. I deffinitely had that in mind when I was thinking through everything, however I've been thinking about it on my own for some time and was anxious to get the question up here so I could get an answer
One basic idea would be this: Start with a function like where is large enough to make duck under function Then make large enough to get you to the x axis faster than does. Then glue the zero function onto this much. So you have this:
Here is defined by