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

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

Answer 1

a^+b^2≥2ab,b^2+c^2≥2bc,c^2+a^2≥2ca,
相加,除以2,得：a^2+b^2+c^2≥ab+bc+ca.
由柯西不等式,
(bc+ca+ab)[a^4/(bc)+b^4/(ca)+c^4/(ab)]≥(a^2+b^2+c^2)^2,
a^4/(bc)+b^4/(ca)+c^4/(ab)≥(a^2+b^2+c^2)^2/(bc+ca+ab)≥a^2+b^2+c^2,
a^5+b^5+c^5≥a^3bc+b^3ac+c^3ab.