若\(a+b+c=1\),用廣義科西就很簡單了
\( \displaystyle \left[\frac{1}{a}\left( \frac{1}{1+b}\right)+\frac{1}{b}\left( \frac{1}{1+c}\right)+\frac{1}{c}\left( \frac{1}{1+a}\right) \right] \left[(1+b)+(1+c)+(1+a) \right] \left[ a+b+c \right] \ge (1+1+1)^3\)
\( \displaystyle \left[ \frac{1}{a}\left( \frac{1}{1+b}\right)+\frac{1}{b}\left( \frac{1}{1+c}\right)+\frac{1}{c}\left( \frac{1}{1+a}\right) \right] \ge \frac{27}{4} \)
