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 composition of the increasing convex function exp  and the convex function log f, but the converse is not always true. For example f(x) = x² is a convex function, but log f(x) = log x² = 2log  | x | is not a convex function and thus f(x) = x² 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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