Assume that we have a concave and increasing function f(x). Consider some positive value L>0 and the difference of this function in two points: $f(x+L)-f(x)$. Can we prove the following relation of the following logarithmic derivatives: $\frac{f''(x)}{f'(x)}<\frac{f'(x+L)-f'(x)}{f(x+L)-f(x)}$? It is possible to assume that $f'(x)$ is completely monotone (if such condition is necessary).
Some ideas:
- I tried to use Taylor expansion for f(x+L) around f(x). Considering the difference between these two logarithmic derivatives, I have got that:
$\sum_{n=2}^{\infty} \frac{L^n}{n!} \left[ f^{(n+1)}f^{(1)}-f^{(n)}f^{(2)} \right]$. I do not know what to do next...
If we assume f'(x) to be completely monotone, then, it is possible to prove that $|f^{(n+1)}f^{(1)}|>|f^{(n)}f^{(2)}|$ (due to Schur-convexity of f'(x)). This means that the first component (for n=2) is positive, but the next one will be negative (and so on). This does not allow me to make a decision if the difference is clearly positive or negative. - Another idea was to consider $f(x+L)-f(x)$ as some function $g(x)$ (that also should be completely monotone), but then I do not know how to proceed.
Any help? Any counterexample?