设a,b,c为正实数,求证；a^5+b^5+c^5大于等于a^3bc+b^3ac+c^3ab急

Question

设a,b,c为正实数,求证；a^5+b^5+c^5大于等于a^3bc+b^3ac+c^3ab
急

Answer 1

a^5+b^5+c^5>=a^3bc+b^3ac+c^3ab
即证a^4/bc+b^4/ac+c^4/bc>=a^2+b^2+c^2
(ab+bc+ac)(a^4/bc+b^4/ac+c^4/bc)
=a^4+b^4+c^4+b^5/a+a^5/b+c^5/b+b^5/c+c^5/a+a^5/c
>=a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2
=(a^2+b^2+c^2)^2
即a^4/bc+b^4/ac+c^4/bc>=(a^2+b^2+c^2)^2/(ab+bc+ac)
又ab+bc+ac=(a^2+b^2+c^2)^2/(ab+bc+ac)>=(a^2+b^2+c^2)^2/(a^2+b^2+c^2)=a^2+b^2+c^2
原不等式得证
(ab+bc+ac)(a^4/bc+b^4/ac+c^4/bc)>=(a^2+b^2+c^2)^2亦可直接由柯西不等式得到