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单词 ProofOfRearrangementInequality
释义

proof of rearrangement inequality


We first prove the rearrangement inequality for the case n=2. Letx1,x2,y1,y2 be real numbers such that x1x2 and y1y2. Then

(x2-x1)(y2-y1)0,

and therefore

x1y1+x2y2x1y2+x2y1.

Equality holds iff x1=x2 or y1=y2.

For the general case, let x1,x2,,xn andy1,y2,,yn be real numbers such that x1x2xn. Suppose that (z1,z2,,zn) is a permutation(rearrangement) of {y1,y2,,yn} such that the sum

x1z1+x2z2++xnzn

is maximized. If there exists a pair i<j with zi>zj, thenxizj+xjzixizi+xjzj (the n=2 case); equality holds iffxi=xj. Therefore, x1z1+x2z2++xnzn is not maximalunless z1z2zn or xi=xj for all pairs i<jsuch that zi>zj. In the latter case, we can consecutivelyinterchange these pairs until z1z2zn (this ispossible because the number of pairs i<j with zi>zj decreaseswith each step). So x1z1+x2z2++xnzn is maximized if

z1z2zn.

To show that x1z1+x2z2++xnzn is minimal for apermutation (z1,z2,,zn) of {y1,y2,,yn} ifz1z2zn, observe that-(x1z1+x2z2++xnzn)=x1(-z1)+x2(-z2)++xn(-zn)is maximized if -z1-z2-zn. This implies thatx1z1+x2z2++xnzn is minimized if

z1z2zn.
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