# Can the real part of an entire function be bounded above by a polynomial? [duplicate]

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function such that $Re(f)\le |p(z)|$ for some polynomial, can we derive that $f(z)$ is a polynomial.
If $p(z)$ is constant, then this can be shown by considering $e^f$. If we instead consider $|u(z)|\le |p(z)|$, then it can also be shown. But if we do not establish the lowerbound, then I cannot figure out how to generlize the proof.
• What is the absolute value of $\exp if(z)$? – Mariano Suárez-Álvarez Feb 9 '16 at 7:29
• you didn't say on what set $u(z) \le p(z)$ it is not the same if it is on a line, on a disk, on the whole complex plane – reuns Feb 9 '16 at 7:33