Inequality $|z_1+z_2|^2 \le (1+|z_1|^2)(1+|z_2|^2)$ I have a problem to prove this inequality 
$|z_1+z_2|^2 \le (1+|z_1|^2)(1+|z_2|^2)$ $\forall (z_1, z_2)\in \mathbb{C}$.
I tried to take the right hand set and subtract the lfs and after simplification I got this:
$1+(ax)^2+(by)^2 -2(ax+by)+(ay)^2+(bx)^2$  and I couldn't prove thqt this result is positive.
Any help please?

$$
(1+|z_1|^2)(1+|z_2|^2)=|z_1|^2+|z_2|^2+1+|z_1|^2|z_2|^2
$$
Now using AM-GM inequality we have:
$$
1+|z_1|^2|z_2|^2\geq 2|z_1||z_2|
$$
Hence
$$
(1+|z_1|^2)(1+|z_2|^2)\geq |z_1|^2+|z_2|^2+ 2|z_1||z_2|= (|z_1|+|z_2|)^2 \ge |z_1+z_2|^2
$$
