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 if $\log f$ 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 convex function $\log f$, 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. An important example of a logarithmically convex function is the gamma function on the positive reals (see also the Bohr–Mollerup theorem).

References

• John B. Conway. Functions of One Complex Variable I, second edition. Springer-Verlag, 1995. ISBN 0-387-90328-3.

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