已知a、b、c是不全相等的三个正数,求证:1/a+1/b+1/c>1/根号ab+1/根号bc+1/根号ac

Question

已知a、b、c是不全相等的三个正数,求证:1/a+1/b+1/c>1/根号ab+1/根号bc+1/根号ac

Answer 1

1/a+1/b+1/c=(1/a+1/b)/2+(1/b+1/c)/2+(1/c+1/a)/2>1/√(ab)+1/√(bc)+1/(ac)
最后一步用到了均值不等式,取严格的大于号是因为a,b,c不全相等.