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 , the composition of the logarithmic function with f, is a convex function.
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function and the function that is supposed convex, but the converse is not always true. For example is a convex function, but is not a convex function and thus is not logarithmically convex. On the other hand, is logarithmically convex since is convex. An important example of a logarithmically convex function is the gamma function on the positive reals (see also the Bohr–Mollerup theorem).
- Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.
- John B. Conway. Functions of One Complex Variable I, second edition. Springer-Verlag, 1995. ISBN 0-387-90328-3.