In complex analysis, Hurwitz's  theorem  roughly states that, under certain conditions, if a sequence of holomorphic functions converges uniformly to a holomorphic function on compact sets, then after a while  (above a certain rank ) those functions and the limit function have the same number of zeros  in any open disk.


I read somewhere that the result is valid more generally on bounded convex sets , from what I remember, but no references were given.


Question. Where can I find this more general statement of Hurwitz's theorem, and where can I find some useful references in this direction? Thank you.



What is the most general form of Hurwitz's theorem in complex analysis?
This has nothing to do with convexity. The exact formulation is this:


Let $\Omega$ be an arbitrary region, and $f\_n\to f$ is a sequence of holomorphic functions converging uniformly on compacts in $\Omega$.
If $f\neq 0$ (this is an important condition!), and $D$ is a region,
such that $\overline{D}\subset \Omega$, and $f(z)\neq 0$ on $\partial D\_1$, then there exists $N$ (dependng on $D$) such that for $n\geq N$, $f$ and $f\_n$ have the same number of zeros
in $D$.


For the proof, you find an intermediate region $D\_1$ such that
$D\subset D\_1\subset\overline{D\_1}\subset\Omega$, and $\partial D\_1$ is piecewise smooth, and the number of zeros of $f$ in $\overline{D\_1}$ is the same as the number of zeros in $D$. Such region exists since $\overline{D}\subset\Omega$, and since the zeros
of $f$ are isolated and $f(z)\neq 0,\; z\in\partial D$.
Then the number
of zeros $f$ in $D$ is the same as in $D\_1$ and is equal to
$$\frac{1}{2\pi i}\int\_{\partial D\_1} \frac{df}{f},$$
and the number of zeros of $f\_n$ in $D\_1$ is
$$\frac{1}{2\pi i}\int\_{\partial D\_1} \frac{df\_n}{f\_n}.$$
Since $f\_n\to f$ uniformly on the compact set $K:=\overline{D\_1}\backslash D$, and $f(z)\neq 0$ on $K$, we conclude that for $n\geq N$, $f\_n$ has the same number of zeros
in $D$ and $D\_1$. Since the integrals for $f\_n^\prime/f\_n$ converge to the
integral for $f'/f$, and all these integrals are integers, we conclude that they are equal for $n\geq N$.


Refs. This is stated in full generality in the book


A. I. Markushevich, Theory of functions of a complex variable,
vol. I, Ch. IV, Sect 3 (p. 426 of the Russian original).


Special cases are in:


Ahlfors, Complex Analysis, p. 178, Theorem 2,


Titchmarsh, The theory of functions, sect. 3.45,


Marshall, Complex Analysis, Theorem 8.8,


But all these special cases have the same proof as the general theorem whose complete proof I wrote.


