若x1,x2,x3.xn属于正实数,求证：x1^x1*x2^x2.*xn^xn>=(x1*x2...*xn)^((x1+x2...+xn)/n)

Question

若x1,x2,x3.xn属于正实数,求证：x1^x1*x2^x2.*xn^xn>=(x1*x2...*xn)^((x1+x2...+xn)/n)

Answer 1

证明：
利用均值不等式a+b≥2√ab,可得
x1^2/(1+x1)+(1+x1)/(n+1)^2≥2√[(1+x1)/(n+1)^2*(x1^2/(1+x1)]=2x1/(n+1)
x2^2/(1+x2)+(1+x2)/(n+1)^2≥2√[1+x2)/(n+1)^2*(x2^2/(1+x2)]=2x2/(n+1)
……………………
xn^2/(1+xn)+(1+xn)/(n+1)^2≥2√[1+xn)/(n+1)^2*(xn^2/(1+xn)]=2xn/(n+1)
以上各不等式相加,可得
x1^2/(1+x1)+x2^2/(1+x2)+……+xn^2/(1+xn)+(x1+x2+……+xn+n)/(n+1)^2
≥2(x1+x2+……+xn)/(n+1)
因为x1+x2+...+xn=1,则不等式整理,可得
x1^2/(1+x1)+x2^2/(1+x2)+……+xn^2/(1+xn)+1/(n+1)≥2/(n+1),即
x1^2/(1+x1)+x2^2/(1+x2)+……+xn^2/(1+xn)≥1/(n+1)