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In [[mathematics]], a [[function (mathematics)|function]] ''f'' defined on a [[convex set|convex subset]] of a [[real numbers|real]] [[vector space]] and taking [[negative and positive numbers|positive]] values is said to be '''logarithmically convex''' or '''superconvex'''<ref>Kingman, J.F.C.  1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.</ref> if <math>{\log}\circ f</math>, the [[function composition|composition]] of the [[logarithmic function]] with ''f'', is a [[convex function]].
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A logarithmically convex function ''f'' is a convex function since it is the [[function composition|composite]] of the [[increasing function|increasing]] convex function <math>\exp</math> and the function <math>\log\circ f</math> that is supposed convex, but the converse is not always true. For example <math>g: x\mapsto x^2</math> is a convex function, but <math>{\log}\circ g: x\mapsto \log x^2 = 2 \log |x|</math> is not a convex function and thus <math>g</math> is not logarithmically convex. On the other hand, <math>x\mapsto e^{x^2}</math> is logarithmically convex since <math>x\mapsto \log e^{x^2} = x^2</math> 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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{{Reflist}}
* John B. Conway. ''Functions of One Complex Variable I'', second edition. Springer-Verlag, 1995. ISBN 0-387-90328-3.


{{PlanetMath attribution|id=5664|title=logarithmically convex function}}
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