$f$ be analytic such that $\Re(f)\ge 0$ then
$\Im(f)=$ constant
$\Im(f)\ge 0$
$f=$ constant
$\Re(f)=|z|$
what I have done is if $f=u+iv$ then consider, $g(z)=e^{u+iv}$ then $|g|=e^{u}$ as $g$ never vanishes so $g$ must be constant so 3 is correct in my guess. 2 is not necesarily true also 4, please help for rigoriousness.