On the existence of limx→0+log(f(x))log(x) \lim_{x \to 0^{+}} \frac{\log(f(x))}{\log(x)} under certain constraints.

I am considering a continuously differentiable real-valued function f:(0,1)→(0,∞) f: (0,1) \to (0,\infty) such that:

f f is decreasing on (0,1) (0,1) .
limx→0+f(x)=∞ \displaystyle \lim_{x \to 0^{+}} f(x) = \infty .
The map x↦x2f′(x) x \mapsto x^{2} f'(x) is decreasing on (0,1) (0,1) .
limx→0+x2f′(x)=0 \displaystyle \lim_{x \to 0^{+}} x^{2} f'(x) = 0 .

I am wondering if, under these constraints, the limit limx→0+log(f(x))log(x) \displaystyle \lim_{x \to 0^{+}} \frac{\log(f(x))}{\log(x)} exists.

Thanks a lot to anybody who has any thought/counterexample, or spent time reading this question!




Thanks for pointing this, I realized I made a sign mistake when writing the question, I meant x↦−x2f(x)x \mapsto -x^2 f(x) is increasing.
– Aurelien
2 days ago



I asked the question on mathoverflow too and it has been solved: mathoverflow.net/questions/252778/…
– Aurelien