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. In effect the logarithm drastically slows down the growth of the original function , so if the composition still retains the convexity property, this must mean that the original function was 'really convex' to begin with, hence the term superconvex.
A logarithmically convex function f is a convex function since it is the composite of the increasing convex function and the function , which is supposed convex. 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).
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- Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. ISBN 9780521833783.
- John B. Conway. Functions of One Complex Variable I, second edition. Springer-Verlag, 1995. ISBN 0-387-90328-3.