## convex functions from surfaces of revolution

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Li-Tam (Ann of math 1987) proved some estimates on nonnegative harmonic functions on manifolds with sec $\geq 0$ outside a compact set. I just want to get some feeling on how nontivial those estimates are. so I tried surface of revolution with positive curvature. In this simple case, those estimates are reduced the following question on convex functions from the graph of the surface.

Assume that $f$ is a smooth convex function on $[0,+\infty]$ with $f(0)=f^{\prime}(0)=0$ and $\lim_{x \rightarrow +\infty} f(x)=\lim_{x \rightarrow +\infty} f^{\prime}(x)=+\infty$ ($f^{\prime}$ is stricty increasing),

Then can we show:

$$\lim_{x \rightarrow +\infty} \frac{\log[\int_{C}^{x} \frac{f^{\prime}(\tau)}{\tau} d\tau \int_{C}^{x} \tau f^{\prime} (\tau) d\tau]} {\log[f(x)]}=2$$ for any fixed number $C>0$?

At first I thought it was just a simple calculus and $\liminf \geq 2$ by Cauchy-Schwarz, but I tried and failed to get the $\limsup$ part. I can only show that $\limsup \leq 2$ when $\lim_{x \rightarrow +\infty} \frac{f(x)}{xf^{\prime}(x)}$ exists. However, this assumption seems not true in general (although I don't have a counterexample yet). I will greatly appreciate if anyone suggests a solution.

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Maybe the computation is made simpler if we change variable, that makes the limit in terms of the inverse $g$ of $f$: $$\lim_{x\to\infty}\frac{\log\left(\int_c^xg(s)ds\int_c^x\frac{1}{g(s)}ds\right)}{\log x}=2\\ .$$  – Pietro Majer May 25 at 22:06
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