In mathematics, a function f is logarithmically convex or superconvex if, the composition of the logarithm with f, is itself a convex function.

Definition

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 \log 0 as -\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 .

Equivalent conditions

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 , Moreover, f is strictly logarithmically convex if and only if these inequalities are always strict. If f is twice differentiable, then it is logarithmically convex if and only if, for all x in I , If the inequality is always strict, then f is strictly logarithmically convex. However, the converse is false: It is possible that f is strictly logarithmically convex and that, for some x , we have. For example, if, then f is strictly logarithmically convex, but. Furthermore, is logarithmically convex if and only if is convex for all.

Sufficient conditions

If are logarithmically convex, and if are non-negative real numbers, then is logarithmically convex. If is any family of logarithmically convex functions, then is logarithmically convex. If is convex and is logarithmically convex and non-decreasing, then g \circ f is logarithmically convex.

Properties

A logarithmically convex function f is a convex function since it is the composite of the increasing convex function \exp and the function \log\circ f, which is by definition convex. However, being logarithmically convex is a strictly stronger property than being convex. For example, the squaring function f(x) = x^2 is convex, but its logarithm is not. Therefore the squaring function is not logarithmically convex.

Examples