请输入您要查询的字词:

 

单词 ProofOfGeneralMeansInequality
释义

proof of general means inequality


Let w1, w2, …, wn be positive real numbers such thatw1+w2++wn=1. For any real number r0, theweighted power mean of degree r of n positive real numbers x1,x2, …, xn (with respect to the weights w1, …,wn) is defined as

Mwr(x1,x2,,xn)=(w1x1r+w2x2r++wnxnr)1/r.

The definition is extended to the case r=0 by taking the limitr0; this yields the weighted geometric mean

Mw0(x1,x2,,xn)=x1w1x2w2xnwn

(see derivation of zeroth weighted power mean). We will prove theweighted power means inequalityMathworldPlanetmath, which states that for any two realnumbers r<s, the weighted power means of orders r and sof n positive real numbers x1, x2, …, xn satisfy theinequality

Mwr(x1,x2,,xn)Mws(x1,x2,,xn)

with equality if and only if all the xi are equal.

First, let us suppose that r and s are nonzero. Wedistinguish three cases for the signs of r and s: r<s<0,r<0<s, and 0<r<s. Let us consider the last case, i.e. assumer and s are both positive; the others are similar. We writet=sr and yi=xir for 1in; this impliesyit=xis. Consider the function

f:(0,)(0,)
xxt.

Since t>1, the second derivative of f satisfiesf′′(x)=t(t-1)xt-2>0 for all x>0, so f is a strictlyconvex function. Therefore, according to Jensen’s inequality,

(w1y1+w2y2++wnyn)t=f(w1y1+w2y2++wnyn)
w1f(y1)+w2f(y2)++wnf(yn)
=w1y1t+w2y2t++wnynt,

with equality if and only if y1=y2==yn. By substitutingt=sr and yi=xir back into this inequality, we get

(w1x1r+w2x2r++wnxnr)s/rw1x1s+w2x2s++wnxns

with equality if and only if x1=x2==xn. Since s ispositive, the function xx1/s is strictly increasing, soraising both sides to the power 1/s preserves the inequality:

(w1x1r+w2x2r++wnxnr)1/r(w1x1s+w2x2s++wnxns)1/s,

which is the inequality we had to prove. Equality holds if and onlyif all the xi are equal.

If r=0, the inequality is still correct: Mw0 is defined aslimr0Mwr, and since MwrMws for all r<s withr0, the same holds for the limit r0. The same argumentMathworldPlanetmathshows that the inequality also holds for s=0, i.e. thatMwrMw0 for all r<0. We conclude that for all real numbersr and s such that r<s,

Mwr(x1,x2,,xn)Mws(x1,x2,,xn).
Titleproof of general means inequality
Canonical nameProofOfGeneralMeansInequality
Date of creation2013-03-22 13:10:26
Last modified on2013-03-22 13:10:26
Ownerpbruin (1001)
Last modified bypbruin (1001)
Numerical id5
Authorpbruin (1001)
Entry typeProof
Classificationmsc 26D15
Related topicArithmeticMean
Related topicGeometricMean
Related topicHarmonicMean
Related topicRootMeanSquare3
Related topicPowerMean
Related topicWeightedPowerMean
Related topicArithmeticGeometricMeansInequality
Related topicJensensInequality
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/3 22:00:12