Let $f$ be entire and suppose that $Re f(z) \le M$ for all $z$. Prove that $f$ must be a constant function.

Question

Let $f$ be entire and suppose that $\Re f(z) \le M$ for all $z$. Prove that $f$ must be a constant function.

How would you do this proof? I was thinking about applying Liouville's theorrem to the function $e^f$ but I'm not sure..

Answer 1

With $g(z)=e^{f(z)}$, then $|g(z)|=e^{\text{Re}f(z)}\leq e^{M}$, so Liouville's Theorem says that $g=c$, the result follows.