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 $\max \left({\alpha ,\beta}\right)$, with equality if $\alpha \ne \beta$.

Also see

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