Zeros of analytic functions, with or without multiplicities

The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound on the number of distinct zeros of the polynomial $abc$ in terms of the degrees of $a$, $b$ and $c$. We extend this to general analytic functions living on a reasonable bounded domain $\Omega\subset\mathbb C$, rather than on the whole of $\mathbb C$. The estimates obtained are sharp, for any $\Omega$, and a generalization of the original result on polynomials can be recovered from them by a limiting argument.