I am doing some research in information theory related to the $f$-divergences and some of their properties. So we have a convex function $f:(0,\infty)\rightarrow \mathbb R$ such that $f(1)=0$, and from that we can deduce a lot of interesting properties regarding the associated $f$-divergence but in the end it would become handy to be able to have, for any $a$ and $b$ in $(0,\infty)$ $$f(ab)\geq b f(a) + a f(b)$$
And I'm not sure if this is a known property and/or if it is incompatible with the convexity of $f$ and the fact that $f(1)=0$.
Observe that this is true with equality for the Kullback–Leibler divergence as $f(t)=t\log(t)$ and $f(ab)=ab\log(ab)=ab\log(a)+ab\log(b)=b f(a)+a f(b)$
So I'm wondering if some of you have already seen something of this type or related, it would be of help.
From the given relation,
$$\frac{f(ab)}{ab}\ge \frac{f(a)}a+\frac{f(b)}b,$$
which is in the form
$$g(ab)\ge g(a)+g(b),$$
or with $h(t):=g(e^t)$,
$$h(a+b)\ge h(a)+h(b).$$