Possible Duplicate:
$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant
$f$ is entire such that $|\mathrm{Re}f(z)|<|\mathrm{Im}\,f(z)|$ for all $z \in \mathbb C$. To prove $f$ is a constant function.
I know I can prove $f$ do not have essential singularity at $\infty$ for this case which will eventually lead function to be polynomial. Then I can even go further with $f(z)= \alpha z^n $ for some large $z$. Then I don't know where to go. Am I going somewhere with that or just doing something not important. Any hint much appreciated.