In mathematics, a function f is logarithmically convex or superconvex if ${\displaystyle {\log }\circ f}$, the composition of the logarithm with f, is itself a convex function.

Let X be a convex subset of a real vector space, and let f : X → R be a function taking non-negative values. Then f is:

Here we interpret ${\displaystyle \log 0}$ as ${\displaystyle -\infty }$.

Explicitly, f is logarithmically convex if and only if, for all x₁, x₂ ∈ X and all t ∈ [0, 1], the two following equivalent conditions hold:

Similarly, f is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all t ∈ (0, 1).

The above definition permits f to be zero, but if f is logarithmically convex and vanishes anywhere in X, then it vanishes everywhere in the interior of X.

If f is a differentiable function defined on an interval I ⊆ R, then f is logarithmically convex if and only if the following condition holds for all x and y in I:

This is equivalent to the condition that, whenever x and y are in I and x > y,