Complex Analysis Proof. Entire Function.

Question

Let $g(w)$ be an entire function s.t. $$ (1+ |w|^l)^{-1}g^{(m)}(w)$$ is bounded for some natural numbers $l$ and $m$. Prove that $g^{(n)}(w)$ is identically zero for sufficiently large $n$. How large must $n$ be in terms of $l$ and $m$?

Answer 1

From the Cauchy integral formula for function value and derivatives on increasingly large circles you obtain the result that an entire function that is polynomially bounded, $|f(z)|\le p(|z|)$, is in fact itself a polynomial. The degree is bounded by the degree of the bound, $\deg f\le\deg p$.

If the derivative of an entire function is polynomial, then the function itself must also be a polynomial.