Given positive real numbers $a, b, c$ with $aI am trying to prove the following:
$$\frac{a}{1+a} < \frac{b}{1+b} + \frac{c}{1+c}$$
given that $a, b, c > 0$ and $a < b+c$. I tried various rearrangements but can't seem to get anywhere with it.

By C-S
$$\frac{b}{1+b}+\frac{c}{c+1}=\frac{b^2}{b^2+b}+\frac{c^2}{c^2+c}\geq\frac{(b+c)^2}{b^2+c^2+b+c}\geq\frac{(b+c)^2}{(b+c)^2+b+c}=$$
$$=\frac{b+c}{b+c+1}=1-\frac{1}{b+c+1}>1-\frac{1}{a+1}=\frac{a}{a+1}.$$
Done!
