Definition:Order of Entire Function/Definition 2

Definition
Let $f: \C \to \C$ be an entire function.

Let $f$ be not identically zero.

The order $\alpha\in[0,+\infty]$ of $f$ is the infimum of the $\beta\geq 0$ for which
 * $\displaystyle \log\left(\max_{|z|\leq R}|f(z)|\right) = O(R^\beta)$

or $\infty$ if no such $\beta$ exists, where $\mathcal O$ denotes big-O notation

The order of $0$ is $0$.