Help me please.
Let $\displaystyle f(z)$be an entire function :
$\displaystyle f(z)\notin \mathbb{R}, \forall z\in \mathbb{C}$.
Prove that$\displaystyle f $is constant. (Don't use Picard Theorem).
Thanks!
Hint: find a mobius map g that maps the real line to the unit circle C, then the domain of g(f), D={g(f(z))} has no intersection with C. Since D is connected, D is either inside C or outside C, then {1/g(f(z))} is inside C. Anyway this contradicts the maximal module principle.