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I have not even understood the statement clearly to attempt it !

Suppose that $f$ is an entire function and that there exist two real numbers $M > 0$ and $p ≥ 1$ such

that $|f (z)| ≤ M (1 + |z|^p )\quad \forall z \in \mathbb{C}$. Describe, giving a rigorous argument, all the entire functions that satisfy this growth estimate.

I know what an entire function is and Power series development of homomorphic functions. BUt I am unable to understand this.

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replace $p$ by $\lceil p \rceil$ so now $p \in \mathbb{N}$. let $$f_0(z) = f(z), \qquad \qquad \qquad f_{n+1}(z) = \frac{f_n(z)-f_n(0)}{z}$$

hence at least when $|z| \to \infty$ : $f_n(z) \le (M+\epsilon)(1+|z|^{p-n})$ and we can apply the Liouville theorem to $f_p(z)$ which is a bounded entire function. thus $f_p(z) = C$ and

$$f_{n}(z) = z f_{n+1}(z)+f_{n}(0) \quad \text{is a degree $p-n$ polynomial }$$

$\implies f(z) = f_0(z)$ is a degree $p$ polynomial.

(of course if at first $p$ wasn’t an integer the coefficient of $z^{\lceil p \rceil}$ has to be $0$ and $f(z)$ is a degree $\lfloor p \rfloor$ polynomial)

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