I want to prove the following theorem
Let $f$ be an entire function. Suppose that $Re[f(z)] \leq M$, $z \in \mathbb{C}$ for some $M\in \mathbb{R}$. Then $f$ is a constant.
My trial is as follows :
From the Borel–Carathéodory theorem wiki and plugging $R=2r$ with $Re[f(z)] \leq M$, and from this theorem
If $f$ is an entire function satisfying $|f(z)| \leq A + B |z|^k$ for some positive constants $A,B$ and some non-negative integer $k$, then $f$ is a polynomial degree at most $k$
I notice $f$ is a constant function. But I want to prove this without using the Borel–Carathéodory theorem. [Actually, If possible I want to extend the problem $Re[f(z)] \leq M r^\lambda$ with a non-negative integer $\gamma$, and showing that $f$ is a polynomial degree at most $\gamma$. From the Borel–Carathéodory theorem I know this is true.]
