$f$ and $g$ are entire function such that $|f^2+g^2|=1$ Then which of the following are correct?
$f$ and $g$ are constant.
$f$ and $g$ are bounded.
$f$ and $g$ have no zeroes on unit circle.
Well What I do is let $h(z)= f^2(z) + g^2(z)$ then clearly $h(z)$ is bounded entire hence constant by Liouville theorem, hence (4) is correct right?