Logarithmically convex function

In mathematics, a function f defined on an convex subset of a real vector space and taking positive values is said to be logarithmically convex if logf(x) is a convex function of x.

It is easy to see that a logarithmically convex function is a convex function, but the converse is not always true. For example f(x) = x² is a convex function, but logf(x) = logx² = 2log | x | is not a convex function and thus f(x) = x² is not logarithmically convex. On the other hand, is logarithmically convex since 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).

References

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

• This article incorporates material from logarithmically convex function on PlanetMath, which is licensed under the GFDL.