Real part bound

Question

If $f$ is an entire function on $\mathbb{C}$ such that $|\operatorname{Re}(f(z))|\leq k|z|^m$, for all $|z|$ sufficiently large, for fixed positive real $k$ and natural number $m$, then $f$ is a polynomial of degree not exceeding $m$.

Answer 1

One approach is to write $f(z)=\sum_{n=0}^{\infty}a_nz^n$ and then recall

$$\tag 1 \text {Re } f(z) = \frac{1}{2}\left (\sum_{n=0}^{\infty}a_nz^n + \overline {\sum_{n=0}^{\infty}a_nz^n} \right ).$$

Now let $z=re^{it}$ and consider

$$\frac{1}{2\pi}\int_0^{2\pi}|\text {Re } f(re^{it})|^2\,dt.$$

Using $(1)$ and Parseval's theorem then leads to

$$\frac{1}{2}\sum_{n=0}^{\infty}|a_n|^2r^{2n} \le k^2r^{2m}$$

for large $r.$ This implies the desired result.