Related to this question.

For $x_+ \in (0,\infty)$, $a \in \mathbb{R}$ let $F\colon[0,x_+] \to [a,\infty)$ be a twice continuous differentiable (in $(0,x_+)$) function with $f := F'$, $f(x) > 0$, and $f'(x) < 0$ for all $x \in (0,x_+]$. Moreover, we assume that $$\lim_{x \to 0} f(x) = \infty$$ and $F(0) = a$ holds.

**The question:** Does this implies that there exists a $\beta \in (0,\infty)$ such that $f(x)f(y) \ge \beta f(xy)$ for all $x,y \in (0,x_+]$.

This special version came in my mind after i analyzed that the counterexample here relies on the fact that the function $f$ is so steep that the primitive integral $F$ has limit $\lim_{x \to 0}F(x) = -\infty$.