In mathematics, a function f defined on a convex subset of a real vector space and taking positive values is said to be logarithmically convex or superconvex if log ∘ f , the composition of the logarithmic function with f, is a convex function. In effect the logarithm drastically slows down the growth of the original function f , so if the composition still retains the convexity property, this must mean that the original function f was 'really convex' to begin with, hence the term superconvex.
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function exp and the function log ∘ f , which is supposed convex. The converse is not always true: for example g : x ↦ x 2 is a convex function, but log ∘ g : x ↦ log x 2 = 2 log | x | is not a convex function and thus g is not logarithmically convex. On the other hand, x ↦ e x 2 is logarithmically convex since x ↦ log e x 2 = x 2 is convex. An important example of a logarithmically convex function is the gamma function on the positive reals (see also the Bohr–Mollerup theorem).
