首先,由Cauchy不等式,(√a+√b+√c)² ≤ (a+b+c)(1+1+1) = 3,得√a+√b+√c ≤ √3.
同样由Cauchy不等式,((√a+√b)+(√b+√c)+(√c+√a))(1/(√a+√b)+1/(√b+√c)+1/(√c+√a)) ≥ (1+1+1)².
即得1/(√a+√b)+1/(√b+√c)+1/(√c+√a) ≥ 9/(2(√a+√b+√c)) ≥ 3√3/2.