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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 Mahler's inequality ： ウィキペディア英語版 Mahler's inequality In mathematics, Mahler's inequality, named after Kurt Mahler, states that the geometric mean of the term-by-term sum of two finite sequences of positive numbers is greater than or equal to the sum of their two separate geometric means: :$\backslash prod\_^n\; (x\_k\; +\; y\_k)^\; \backslash ge\; \backslash prod\_^n\; x\_k^\; +\; \backslash prod\_^n\; y\_k^$ when ''x''''k'', ''y''''k'' > 0 for all ''k''. == Proof == By the inequality of arithmetic and geometric means, we have: :$\backslash prod\_^n\; \backslash left(\backslash right)^\; \backslash le\; \backslash sum\_^n\; ,$ and : $\backslash prod\_^n\; \backslash left(\backslash right)^\; \backslash le\; \backslash sum\_^n\; .$ Hence, :$\backslash prod\_^n\; \backslash left(\backslash right)^\; +\; \backslash prod\_^n\; \backslash left(\backslash right)^\; \backslash le\; n\; =\; 1.$ Clearing denominators then gives the desired result. 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「Mahler's inequality」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.