Order of Product of Entire Function with Polynomial

Theorem
Let $f:\C\to\C$ be an entire function of order $\omega$.

Let $P:\C\to\C$ be a nonzero polynomial.

Then $f\cdot P$ has order $\omega$.

Proof
By Order of Product of Entire Functions and Polynomial has Order Zero, $f\cdot P$ has order at most $\omega$.

By Limit at Infinity of Polynomial, there exist $r,\delta>0$ such that $|P(z)| \geq \delta$ for $|z|\geq r$.

Suppose $\displaystyle \log\left(\max_{|z|\leq R}|f(z)P(z)|\right) = O(R^\beta)$ for some $\beta < \omega$.

By the Maximum Modulus Principle $\max_{|z|\leq R}|f(z)| \leq \frac1\delta \max_{|z|\leq R}|f(z)P(z)|$ for $R\geq r$.

Thus $\displaystyle \log\left(\max_{|z|\leq R}|f(z)|\right) = O(R^\beta)$.

This is a contradiction.

Thus $f\cdot P$ has order $\omega$.

Also see

 * Order of Product of Entire Functions
 * Polynomial has Order Zero