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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 Schur's inequality ： ウィキペディア英語版 Schur's inequality In mathematics, Schur's inequality, named after Issai Schur, establishes that for all non-negative real numbers ''x'', ''y'', ''z'' and a positive number ''t'', :$x^t\; (x-y)(x-z)\; +\; y^t\; (y-z)(y-x)\; +\; z^t\; (z-x)(z-y)\; \backslash ge\; 0$ with equality if and only if ''x = y = z'' or two of them are equal and the other is zero. When ''t'' is an even positive integer, the inequality holds for all real numbers ''x'', ''y'' and ''z''. When $t=1$, the following well-known special case can be derived: :$x^3\; +\; y^3\; +\; z^3\; +\; 3xyz\; \backslash geq\; xy(x+y)\; +\; xz(x+z)\; +\; yz(y+z)$ == Proof == Since the inequality is symmetric in $x,y,z$ we may assume without loss of generality that $x\; \backslash geq\; y\; \backslash geq\; z$. Then the inequality : $(x-y)()+z^t(x-z)(y-z)\; \backslash geq\; 0\backslash ,$ clearly holds, since every term on the left-hand side of the equation is non-negative. This rearranges to Schur's 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「Schur'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.