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 itself a convex function. A logarithmically convex function will grow faster than any exponential in the long run, meaning that any function satisfying

$\lim _{x\to \infty }{\frac {f}{e^{kx}}}=\infty$

independently of the constant k will be logarithmically convex.

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. 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).