logarithmically convex function

Definition.

A function $f:[a,b]\to ℝ$ such that $f(x)>0$ for all $x$ is saidto be logarithmically convex if $\log f(x)$ is a convex function.

It is easy to see that a logarithmically convex function is a convex function, but the converse is not 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 $e^{x^2}$ 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.

The definition is easily extended to functions $f:U\subset ℝ\to ℝ$, for any connected set $U$ (where still we have $f>0$), in the obvious way. Such a function is logarithmically convex if it is logarithmically convex on all intervals$[a,b]\subset U$.