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
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Let $\map f z = \map \OO {e^{\cmod z^m} }$ and $\map g z = \map \OO {e^{\cmod z^k} }$.
Without loss of generality, 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 $\alpha$.
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