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. In effect the logarithm drastically slows down the growth of the original function f, so if the composition still retains the convexity property, this must mean that the original function f 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 \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.