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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 Rearrangement inequality ： ウィキペディア英語版 Rearrangement inequality In mathematics, the rearrangement inequality〔, Section 10.2, Theorem 368〕 states that :$x\_ny\_1\; +\; \backslash cdots\; +\; x\_1y\_n$ \le x_y_1 + \cdots + x_y_n \le x_1y_1 + \cdots + x_ny_n for every choice of real numbers :$x\_1\backslash le\backslash cdots\backslash le\; x\_n\backslash quad\backslash text\backslash quad\; y\_1\backslash le\backslash cdots\backslash le\; y\_n$ and every permutation :$x\_,\backslash dots,x\_\backslash ,$ of ''x''1, . . ., ''xn''. If the numbers are different, meaning that : then the lower bound is attained only for the permutation which reverses the order, i.e. σ(''i'') = ''n'' − ''i'' + 1 for all ''i'' = 1, ..., ''n'', and the upper bound is attained only for the identity, i.e. σ(''i'') = ''i'' for all ''i'' = 1, ..., ''n''. Note that the rearrangement inequality makes no assumptions on the signs of the real numbers. ==Applications== Many famous inequalities can be proved by the rearrangement inequality, such as the arithmetic mean – geometric mean inequality, the Cauchy–Schwarz inequality, and Chebyshev's sum inequality. 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「Rearrangement 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.