# Logarithmically convex function

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[1] if ${\log}\circ f$, the composition of the logarithmic function with f, is a convex function.
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function $\exp$ and the function $\log\circ f$ that is supposed convex, but the converse is not always true. For example $g: x\mapsto x^2$ is a convex function, but ${\log}\circ g: x\mapsto \log x^2 = 2 \log |x|$ is not a convex function and thus $g$ is not logarithmically convex. On the other hand, $x\mapsto e^{x^2}$ is logarithmically convex since $x\mapsto \log e^{x^2} = x^2$ is convex. An important example of a logarithmically convex function is the gamma function on the positive reals (see also the Bohr–Mollerup theorem).