ongelijkheid

Er geldt achtereenvolgens $$\begin{array}{rcl} \displaystyle \frac{1+abc}{a\left(b+1\right)}+\frac{1+abc} {b\left(c+1\right)}+\frac{1+abc}{c\left(a+1\right)}&=&\displaystyle \sum_{cyc}\frac{1+a+ab(1+c)}{a(b+1)}-3\\&=& \displaystyle \sum_{cyc}\frac{1+a}{a(1+b)}+\sum_{cyc}\frac{ab(1+c)}{a(1+b)}-3 \\&\buildrel{\text{AM-GM}}\over{\ge}&6\sqrt[6]{1} - 3\\&=&3.\end{array}$$