ProblemsusingCauchy-Schwarz不等式

1、精品资料欢迎下载Problems using Cauchy-Schwarz in equality.Problem 1. (Jack Garfu nkel)For non-n egative nu mbers a,b,c , prove the in equality+c Jo+b + cy/a + b Qb + c y/c +a 4Solutio n.Using the Cauchy-Schwarz in equality, we haveI V .a= V Ja(5a + b + 9c)住；V+ & + 9c) | V- -曲+b丿轻+b + k)丿I士+叭+E9C丿/x=5(a

2、+ /i + c)2Y- -+ b +阿丿Thus, one only n eeds to prove(fl +h + c) V-I 一 ； ：心Problem 2. (Vo Quoc Ba Can)For non-negative numbers a,b,c with sum equal to 1, prove thatSolutio n.Using the Cauchy-Schwarz in equality, we have精品资料欢迎下载f _y_y y!a + i2=V4(7 + 4b+c JIQV=9工 + =4a + 4b+cI a +bAa + 4b+ca(a + &

3、+C)+/J-乙Aa + Ab + c吃心吨止去IThus,onert(i7 +Z? + C)+ &4n + 46 + c121r(n +& + c)We have16辽y + 3003工亍护+19巧工725工为+10920工口 $处yp _ yq- _护砂听砂即_25(46/ + 4ZJ+C)(4Z? + 4c + a)(4c + 4a + b)fA163工/ +11工盯备一石工日饬一石工/加25(4(7 + 4b + e)(4? + 4-C + 7)(4c + 4a +b)A= 1210工柑 + 吨少 +233工叔+118980% 0eyeWe prove工t/ +11工口

4、万一6工n咕一6工/加0QC-工Xb +12工/，一工口沅Q-工0胡-工住漩0eyeeWe haveQT砂厶CJC精品资料欢迎下载Vn2&c = V i- Ibc)2orTherefore the in equality is equivale nt to丄(-b2)1+ 2(ab ac- 2bc)2-2,(口 -b1)(ab + ac - 2bc) 0eyeeye砂O工(用b:-2ab-lac + 4bc)20-砂Last in equality is obviously true so the result follows. Equality does not hold.Probl

5、em 3. (Vo Quoc Ba Can)For non-n egative nu mbers a,b,c, prove thatajb2+4r:+bjc，+4a +c7n2+4fc2QO 3(c + 5乃)15a11+10f7i + 3d2)(a -b)2-Ac0班酹+4)+ / + 5rr=工(，- bXob + g - 2珈)SiortycJ3ii + b + 5(?=3(rt + + r)2区0精品资料欢迎下载whereA 69n4+ (536&18c)d3一(410, +410he + 306c2)o2+ (536iJ 410&2e + 436bc2+ 165c3)

6、fl +165&4-3O6bJc-18&2c2+69tes+45c4Using the assumption fH):(UCit is easy to prove that？:- .The resultfollows, equality holds if and only if Problem 4.For positive nu mbers a,b,c, prove thato4b4c4. a + b + c - r + :-7 +- r -73+&3ft5+R，+ fp 2Solutio n.ab5+bF +m3(A+ fe + r)(5+ feJ+o力+&2

7、c3+cc)We havea2b + b2c5+c ai= (ab + ca)ab +bc2+ca)- abc(a2+c +ab + bc + ca)Ctb + be + CO j j 31,71,、=-(ab +bc +ca +ab + be +ca +abc(a + b十(?)a + b + c-abc(a2+b2+c +ab + bc + ca)Clb + be + C13J32,7, t 2，丁、A丿 *J=-(ab + he +ca +ab +bc +c a)-abc(a +b +c )a + b + c(lb + bC + CCI f 1 . 2 , 1 id 2，. 1 - t

8、t.J2 t 1r、-a +b +c) +ab +bc +ca -abc(a +Zr +c) =JThus, one only n eeds to prove| (a2+b2+c2)2+a2b- +bc ca2abc(a + b + az+b:+ c1)Without loss of gen erality, no rmalize the in equality by 亠JL let1 .ab-bc + ca= abc (0 q V) (iT + fr + cXn5十沪 + 疋)十(加十比十w精品资料欢迎下载Then we have -2727equivale nt-to一._.-27It i

9、s clear that，厂 _、一 一 - 厂J.is an increasing function in r , we should haveIn equality is completely proved. Equality occurs if and only if，； 一 一Problem 5. (Phan Thanh Nam) _ V 3For the non-n egative nu mbers a,b,c with sum equal to 1. Letprove thatyja + k(b-c)1+ Jb+迩 -o)，+Qc：k(a -b) V3Solutio n.Using

10、 the Cauchy Schwarz in equality,护a + -*O+F54|七十拧(1 2叫 * 聞-J(l+(l-2g)(1+ (70The in equality isyf打71a+k(b-c)2a +J 3丿1O + LV3丿耳a + (b-c)2工Y)2(b-FV3J精品资料欢迎下载(A+b(a +lx2(a 4-Zn-c)(1 +方)(止 +cXr+ff)So, we have to provea + b + c(a + &)(d + r)(c+ a)Za(b+ c)+&+(?+ 3ab + 3bc+ 3fo)2- +3-e a -bc-(a + b)(

11、!?+ 0Without loss of generality, assume门一- : then we have工他一时 S-切2, 、HJ_+(b c)(d + i + c) J(a-b)(b-c)( (1a -ha +bc(方十+ & + c);bc(a-b)(a + b)(b-c)(a -b -ab + ac + be)=- i-:-i-3 0b(fl十bc)(lr + ca)(a + c)(b十e)(灯 +h+ c)- - - -+cf? (c + a)(q + b + c)9(2 + J3一g(6q +巧)冬3(2 +巧丄_g(6g +巧)=g(3g 1)屈M0, 1Equality occurs if and only if；一；：or一 一 一and its permutatio nsProblem 6. (Vo Quoc Ba Can)For positive nu mbers a,b,c, prove