In mathematics, a function f is logarithmically convex or superconvex if , the composition of the logarithm with f, is itself a convex function.
Let be a convex subset of a real vector space, and let be a function taking non-negative values. Then is:
Here we interpret as .
Explicitly, is logarithmically convex if and only if, for all and all , the two following equivalent conditions hold:
Similarly, is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all .
The above definition permits to be zero, but if is logarithmically convex and vanishes anywhere in , then it vanishes everywhere in the interior of .
If is a differentiable function defined on an interval , then is logarithmically convex if and only if the following condition holds for all and in :
This is equivalent to the condition that, whenever and are in and ,
Moreover, is strictly logarithmically convex if and only if these inequalities are always strict.
If is twice differentiable, then it is logarithmically convex if and only if, for all in ,
If the inequality is always strict, then is strictly logarithmically convex. However, the converse is false: It is possible that is strictly logarithmically convex and that, for some , we have . For example, if , then is strictly logarithmically convex, but .
Furthermore, is logarithmically convex if and only if is convex for all .
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 is logarithmically convex.
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function and the 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 is not. Therefore the squaring function is not logarithmically convex.