When $f(x)f(y) \le f(xy)$ holds?

Question

This may be a well-known question (however, I could not find a good answer).

Let $f$ be a positive function on $(0,1).$ My question is the following.

Under what conditions on $f$, the inequality $f(x)f(y) \le f(xy)$ holds for any $x,y \in (0,1)$?

If $f(x)=Ax^{\alpha }$ for some $A>0$ and $\alpha \in \mathbb{R}$, then the above inequality trivially holds. But I want to know the "quick" criteria for the above inequality to hold. For example, does it hold if $f$ is concave?

Answer 1

Partial answer: $f(x)=e-e^{x}$ is a positive concave function on $(0,1)$. You can check that the inequality fails for $x=y=\epsilon$ if $\epsilon >0$ is small enough. (Use Taylor expansion).