Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$

Question

Show that $\frac c {1+c} \le \frac a {1+a} + \frac b {1+b}$ , for $c \le a+b$ and $a,b,c \ge 0$

So need to show $\frac c {1+c} \le \frac {a+b+2ab} {1+a+b+ab}$

We have $\frac c {1+c} \le \frac {a+b} {1+c}\le \frac {a+b+2ab} {1+c}$

So the whole thing reduces to showing $c \ge a+b+ab$ ?

Can anyone help me with this, I'm rubbish at inequalities...

Answer 1

Hint $$\frac c {1+c} \le \frac {a+b+2ab} {1+a+b+ab}\longleftrightarrow c+ac+bc+abc\leq a+b+2ab+ac+bc+2abc$$