In mathematics, a function f is logarithmically convex or superconvex[1] if
log
, 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.
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, 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
, then f is strictly logarithmically convex, but
is logarithmically convex if and only if
α x
are logarithmically convex, and if
are non-negative real numbers, then
is logarithmically convex.
is any family of logarithmically convex functions, then
is logarithmically convex.
is logarithmically convex and non-decreasing, then
is logarithmically convex.
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function
, which is by definition convex.
However, being logarithmically convex is a strictly stronger property than being convex.
For example, the squaring function
is convex, but its logarithm
Therefore the squaring function is not logarithmically convex.
This article incorporates material from logarithmically convex function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.