$\frac{d^2}{dx^2}f(x)>\frac{d^2}{d x^2}g(x) \implies \frac{d^2}{d x^2}\log(f(x))> \frac{d^2}{d x^2}\log(g(x))$?

Question

Does the following inequality hold? $$\frac{d^2}{dx^2}f(x)>\frac{d^2}{d x^2}g(x) \implies \frac{d^2}{d x^2}\log(f(x))> \frac{d^2}{d x^2}\log(g(x))$$

EDIT: as this does not hold i wonder whether it works with $\geq$ ?

To give some insights why I was interested in this question:

The fisher-information is defines as $$ I(\theta)=\int \frac{\partial^2}{\partial \theta ^2}log(f_\theta(x))dP_\theta(x)$$ and I was trying to figure out whether this cannot only be interpreted as the "expected curvature" of the log-likelihood but also as containing information about the expected curvature of the likelihood, but the relationship is at least not as simple as I hoped it would be.

Answer 1

Hint: Consider $f(x)=e^{-x}$ and $g(x) = c$, where $c>0$ is a real number.