The uvw method

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1、Theuvwmethod-TejsTheuvwmethodbyMathiasBækTejsKnudsen1Theuvwmethod-Tejs1BASICCONCEPTS1BasicConceptsThebasicconceptofthemethodisthis:Withaninequalityinthenumbersa,b,c∈R,wewriteeverythingintermsof3u=a+b+c,3v2=ab+bc+ca,w3=abc.Theconditiona,b,c≥0isgiveninmanyinequalities.Inthiscas

2、e,weseethatu,v2,w3≥0.Butifthisisnotthecase,u,v2,w3mightbenegative.E.g.a=−1,b=−2,c=3weget3v2=−7.Sodon’tbeconfused!Sometimesv2canbenegative!TheIdiot-Theorem:u≥v≥w,whena,b,c≥0.PProof:9u2−9v2=a2+b2+c2−ab−bc−ca=1(a−b)2≥0.Andhenceu≥v.2cyc√FromtheAM-GMinequalityweseethatv2=ab+bc+ca≥

3、3a2b2c2=w2,andhencev≥w,3whichconcludesu≥v≥w.Theidiot-theoremisusuallyoflittleusealone,butitisimportanttoknow.Sometimesweintroducetheratiot=u2:v2.Ifu>0,v=0wedefinet=∞,andifu<0,v=0wedefinet=−∞.Ifu=v=0wesett=1.Whena,b,c≥0,weseethatt≥1.Andwealwaysseethat

4、t

5、≥1.TheUVW-Theorem:Givenu,

6、v2,w3∈R:∃a,b,c∈Rsuchthat3u=a+b+c,3v2=ab+bc+ca,w3=abcifandonlyif:hppiu2≥v2andw3∈3uv2−2u3−2(u2−v2)3,3uv2−2u3+2(u2−v2)3Proof:Definef(t)=t3−3ut2+3v2t−w3.Leta,b,cbeitsroots.Byviet´esformulasweseethat3u=a+b+c,3v2=ab+bc+ca,w3=abc.Lemma1:a,b,c∈R⇐⇒(a−b)(b−c)(c−a)∈R.Proofoflemma1:It’str

7、ivialtoseethata,b,c∈R⇒(a−b)(b−c)(c−a)∈R.ThenI’llshowa,b,c/∈R⇒(a−b)(b−c)(c−a)∈/R.Letz/∈Rbeacomplexnumbersuchthatf(z)=0.Thenf(z)=f(z)=0=0,soifzisacomplexrootinf(t),soisz.Assumewlogbecauseofsymmetrythata=z,b=z.Then(a−b)(b−c)(c−a)=−(z−z)

8、z−c

9、2∈/R.Thelastfollowsfrom(a−b)(b−c)(c−a)

10、6=0,i(z−z)∈R,and

11、z−c

12、2∈R.It’sobviousthatx∈R⇐⇒x2∈[0;+∞).So∃a,b,c∈Rsuchthat3u=a+b+c,3v3=ab+bc+ca,w3=abcifandonlyif(a−b)2(b−c)2(c−a)2≥0,wherea,b,caretherootsoff(t).But(a−b)2(b−c)2(c−a)2=27(−(w3−(3uv2−2u3))2+4(u2−v2)3)≥0⇐⇒4(u2−v2)3≥(w3−(3uv2−2u3))2.Fromthiswewouldrequireu2≥v2.But

13、4(u2−v2)3≥(w3−(3uv2−2u3))2⇐⇒p2(u2−v2)3≥

14、w3−(3uv2−2u3)

15、Whenw3≥(3uv2−2u3)it’sequivalentto:pw3≤3uv2−2u3+2(u2−v2)3Whenw3≤(3uv2−2u3)it’sequivalentto:pw3≥3uv2−2u3−2(u2−v2)3pSo(a−b)2(b−c)2(c−a)2≥0⇐⇒w3∈[3uv2−2u3;3uv2−2u3+2(u2−v2)3]∩2Theuvwmethod-Tejs2QUALITATIVEESTIMATIONSpp[3uv2−2u3

16、−2(u2−v2)3;3uv2−2u3]⇐⇒w3∈[3uv2−2u3−2(u2−v2)3;3uv2−2u3+p2(u2−v2)3]and