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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: INEQUALITIES INVOLVING THE MEAN AND THE STANDARD DEVIATION OF NONNEGATIVE REAL NUMBERS | INEQUALITIES INVOLVING THE MEAN AND THE STANDARD DEVIATION OF NONNEGATIVE REAL NUMBERS OSCAR ROJO Received 22 December 2005 Revised 18 August 2006 Accepted 21 September 2006 Let m y y n 1 yj n and s y pm y2 - m2 y be the mean and the standard deviation q .q q .q ofthe components ofthe vector y y1 y2 . yn-1 yn where yq yi y2 . yn-1 yn with q a positive integer. Here we prove that if y 0 then m y2p 1 Vn - 1 s y2p ựm y2P 1 1 Vn - 1 s y2p 1 for p 0 1 2 . The equality holds if and only if the n - 1 largest components of y are equal. It follows that l2p y rp 0 l2p y m y2P 1 Vn - 1 s y2p 2 is a strictly increasing sequence converging to y1 the largest component of y except if the n - 1 largest components of y are equal. In this case l2p y y1 for all p. Copyright 2006 Oscar Rojo. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let m x j 1 j s x Jm x2 - m2 x 1.1 n be the mean and the standard deviation of the components of x x1 x2 . xn-1 xn where xq xq xq . xq-1 xq for apositive integer q. The following theorem is due to Wolkowicz and Styan 3 Theorem 2.1. . Theorem 1.1. Let x1 x2 xn-1 xn. 1.2 Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 43465 Pages 1-15 DOI 10.1155 JIA 2006 43465 2 Inequalities on the mean and standard deviation Then m x -j 1 s x X1 X1 m x fn - 1s x . 1.3 1.4 Equality holds in 1.3 if and only ifx1 x2 xn-1. Equality holds in 1.4 if and only if X2 X3 Xn. Let X1 x2 . xn- 1 xn be complex numbers such that X1 is a positive real number and X1 X2 I Xn-1 I Xn . 1.5 Then Xp x2 p I Xn-1 I p xn p 1.6 for any positive integer p. We apply Theorem 1.1 to 1.6 to obtain m x p s x p xp Vn - 1 1.7 x1 m x p fn - 1s x p where x X1 X1 . xn-1 xn . Then 1 1 p Ip x m x p n-ĩ s x p 1.8 is a sequence of lower bounds for X1 and Up x m fx.ự fn - 1s x p 1 p 1.9
