Order of Reciprocal of Entire Function

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

Let $f$ have no zeroes.

Then $1/f$ has order $\rho$.

Proof
By Zerofree Analytic Function on Simply Connected Set has Logarithm, there exists an entire function $g$ with $f = \exp g$.

Also see

 * Zerofree Entire Function of Finite Order is Exponential of Polynomial