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 if $\log f(x)$ is a convex function of $x$ .

A logarithmically convex function $f$ is a convex function since it is the composition of two convex functions, $\exp$ and $\log f$ . But the converse is not always true. For example $f(x)=x^{2}$ is a convex function, but $\log f(x)=\log x^{2}=2\log |x|$ is not a convex function and thus $f(x)=x^{2}$ is not logarithmically convex. On the other hand, $f(x)=e^{x^{2}}$ is logarithmically convex since $\log e^{x^{2}}=x^{2}$ is convex. A less trivial example of a logarithmically convex function is the gamma function, if restricted to the positive reals (see also the Bohr–Mollerup theorem).