Is it true that if $|f(z)| \leq p(|z|)$ for some polynomial $p$, then $f(z)$ is a polynomial?

Question

Suppose that we have an entire function $f(z)$, which is bounded by: $$|f(z)|\leq p(|z|)$$ for some polynomial $p(z)$. Is it possible to prove that $f(z)$ is a polynomial? I know that it is easy for $p(z)=z^n$, but what if we take some general polynomial? Also is there something we can deduce if $p$ is some other entire function?

Answer 1

Use $p(z)=\sum_{k=1}^n a_n z^n \implies p(|z|) \leq K|z|^n \ \forall |z|>R$ for some R, $K=\sum |a_n|$ for general polynomial.