|
In mathematics, there are two different results that share the common name of the Ky Fan inequality. One is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval. The result was published on page 5 of the book ''Inequalities'' by Beckenbach and Bellman (1961), who refer to an unpublished result of Ky Fan. They mention the result in connection with the inequality of arithmetic and geometric means and Augustin Louis Cauchy's proof of this inequality by forward-backward-induction; a method which can also be used to prove the Ky Fan inequality. The Ky Fan inequality is a special case of Levinson's inequality and also the starting point for several generalizations and refinements, some of them are given in the references below. ==Statement of the classical version== If ''xi'' with 0 ≤ ''xi'' ≤ ½ for ''i'' = 1, ..., ''n'' are real numbers, then : with equality if and only if ''x''1 = ''x''2 = . . . = ''xn''. 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「Ky Fan inequality」の詳細全文を読む スポンサード リンク
|