Logarithmically convex function

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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[1] if {\log}\circ f, 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 \exp and the function \log\circ f, which is supposed convex. The converse is not always true: for example g: x\mapsto x^2 is a convex function, but {\log}\circ g: x\mapsto \log x^2 = 2 \log |x| is not a convex function and thus g is not logarithmically convex. On the other hand, x\mapsto e^{x^2} is logarithmically convex since x\mapsto \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[edit]

  1. ^ 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.
  • Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. ISBN 9780521833783.

See also[edit]

This article incorporates material from logarithmically convex function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.