Say $f:\mathbb{R}\to(0,\infty)$ is measurable, and that $$\forall a,b\in\mathbb{R}~~a<b\implies\log f(a)+\log f(b)\leq2\log f(\frac{a+b}{2}).$$ Why must $f$ be log-concave? (That is, why must $$\forall a,b\in\mathbb{R}~\forall t\in(0,1)~~a<b\implies t\log f(a)+(1-t)\log f(b)\leq\log f(ta+(1-t)b)$$ hold?)

I came across this statement as I was reading the proof of Proposition 2.3.a in Saumard and Wellner. I was thinking about doing something with dyadic approximations of real numbers, but I haven't been able to make this idea rigorous yet.

*Saumard, Adrien; Wellner, Jon A.*, **Log-concavity and strong log-concavity: a review**, Stat. Surv. 8, 45-114 (2014). ZBL1360.62055.