Order of Sum of Entire Functions

Theorem
Let $f, g: \C \to \C$ be entire functions of order $\alpha$ and $\beta$.

Then $f + g$ has order at most $\map \max {\alpha, \beta}$, with equality if $\alpha \ne \beta$.

Proof
Let $\map f z = \map \OO {e^{\cmod z^m} }$ and $\map g z = \map \OO {e^{\cmod z^k} }$.

Then:
 * $\map f z + \map g z = \map \OO {e^{\cmod z^m} } + \map \OO {e^{\cmod z^k} }$

, let $m \ge k$.

Then it follows that $\map f z + \map g z = \map \OO {e^{\cmod z^m} }$.

Then the order of $f + g$ is at most $m$.

Since $\alpha, \beta$ are the infima of $m, k$, it follows that the order of $f + g$ is at most $\alpha$.

Let $\alpha > \beta$ and $\map f z = \map \OO {e^{\cmod z^m} }$.

Then:
 * $\map g z = \map {\mathcal o} {e^{\cmod z^m} }$

So $\dfrac {\map g z} {e^{\cmod z^m} } \to 0$ as $\cmod z \to \infty$.

In particular:
 * $\dfrac {\map f z + \map g z} {e^{\cmod z^m} } \sim \dfrac {\map f z} {e^{\cmod z^m} } > 0$

and minimizing $m$ we see that the order of $f + g$ is at least $\alpha$.

Also see

 * Order of Product of Entire Functions
 * Order of Product of Entire Function with Polynomial