请教两道不等式证明题:1、若x,y,z属于R+,且x+y+z=xyz,证明不等式(y+z)/x+
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请教两道不等式证明题:1、若x,y,z属于R+,且x+y+z=xyz,证明不等式(y+z)/x+
请教两道不等式证明题:
1、若x,y,z属于R+,且x+y+z=xyz,证明不等式(y+z)/x+(z+x)/y+(x+y)/z大于等于2(1/x+1/y+1/z)^2.
2、已知0小于等于a,b,c小于等于1,求证:a/(bc+1)+b/(ca+1)+c/(ab+1)小于等于2 .
请教两道不等式证明题:
1、若x,y,z属于R+,且x+y+z=xyz,证明不等式(y+z)/x+(z+x)/y+(x+y)/z大于等于2(1/x+1/y+1/z)^2.
2、已知0小于等于a,b,c小于等于1,求证:a/(bc+1)+b/(ca+1)+c/(ab+1)小于等于2 .
1.不等式等价于xyz(xy(x+y)+yz(y+z)+zx(z+x)) ≥ 2(xy+yz+zx)².
由xyz = x+y+z,进一步等价于(x+y+z)(xy(x+y)+yz(y+z)+zx(z+x)) ≥ 2(xy+yz+zx)².
也即((x+y)+(y+z)+(z+x))(z²(x+y)+x²(y+z)+y²(z+x)) ≥ (z(x+y)+x(y+z)+y(z+x))².
易见这由Cauchy不等式立即得到.
2.由对称性,不妨设a ≤ b ≤ c.
则a/(bc+1)+b/(ca+1) ≤ a/(ab+1)+b/(ab+1) = (a+b)/(ab+1) = 1-(1-a)(1-b)/(ab+1) ≤ 1.
又c/(ab+1) ≤ c ≤ 1.
相加即得a/(bc+1)+b/(ca+1)+c/(ab+1) ≤ 2.
由xyz = x+y+z,进一步等价于(x+y+z)(xy(x+y)+yz(y+z)+zx(z+x)) ≥ 2(xy+yz+zx)².
也即((x+y)+(y+z)+(z+x))(z²(x+y)+x²(y+z)+y²(z+x)) ≥ (z(x+y)+x(y+z)+y(z+x))².
易见这由Cauchy不等式立即得到.
2.由对称性,不妨设a ≤ b ≤ c.
则a/(bc+1)+b/(ca+1) ≤ a/(ab+1)+b/(ab+1) = (a+b)/(ab+1) = 1-(1-a)(1-b)/(ab+1) ≤ 1.
又c/(ab+1) ≤ c ≤ 1.
相加即得a/(bc+1)+b/(ca+1)+c/(ab+1) ≤ 2.
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