Log-concavity

A function $$f:\mathbb R^n \to \mathbb R$$ is logarithmically concave or log-concave if $$f(x) > 0 $$ for all $$x \in \textbf{dom} f$$ and $$\log f$$ is concave. A function is log-convex iff $$1/f$$ is log-concave.

Criterion
Log-concavity can be expressed directly, without logarithms, through the following inequality:

$$f(\theta x+(1-\theta y)) \ge f(x)^\theta f(y)^{(1-\theta)} $$

Operations that preserve log-concavity
Log-convexity and log-concavity are closed under multiplication and positive scaling. Sums of log-concave functions are not, in general, log-concave. However, sums of log-convex functions are log-concave.

Of note, in some special cases, log-concavity is preserved by integration. If $$f:\mathbb R^n \times \mathbb R^m \to \mathbb R$$ is log-concave, then

$$g(x) = \int f(x,y) dy$$

is alog-concave function of $$x$$ (The integration here is over $$\mathbb R^m$$).