For convex $f$, why is $(p,q) \mapsto q \, f(p/q)$ convex on $\mathbb{R}_+^2$?

Question

This fact was stated in the Wikipedia article on $f$-divergences to explain why they are jointly convex.

Answer 1

Actually, I figured it out. You can first prove using Jensen's inequality that for positive $(a_1, a_2, ...)$ and $(b_1, b_2, ...)$

$\sum b_i \, f(a_i/b_i) \geq (\sum b_i) \, f \left( \frac{\sum a_i}{\sum b_i} \right)$

The convexity statement follows.