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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Logarithmically convex function ： ウィキペディア英語版 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〔Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.〕 if $\backslash circ\; f$, 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 $\backslash exp$ and the function $\backslash log\backslash circ\; f$, which is supposed convex. The converse is not always true: for example $g:\; x\backslash mapsto\; x^2$ is a convex function, but $\backslash circ\; g:\; x\backslash mapsto\; \backslash log\; x^2\; =\; 2\; \backslash log\; |x|$ is not a convex function and thus $g$ is not logarithmically convex. On the other hand, $x\backslash mapsto\; e^$ is logarithmically convex since $x\backslash mapsto\; \backslash log\; e^\; =\; x^2$ 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== * 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Logarithmically convex function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.