Say a sequence \(f_k\to f\) locally uniformly on \(\Omega\) if \(f_k\to f\) uniformly on every compact subset. Equivalently, \(f_k \to f\) uniformly on any bounded subset of strictly positive distance to \({{\partial}}\Omega\).
A function \(f:\Omega\to {\mathbf{C}}\) is univalent iff \(f\) is holomorphic and injective. These are exactly the conformal maps of \(\Omega\) to other domains.
Suppose \(\left\{{f_k}\right\}\) is a sequence of holomorphic functions on \(\Omega\) converging locally uniformly to \(f\) on \(\Omega\) and \(f\) has a zero of order \(n\) at \(z_0\). Then there exists some \(\delta\) such that for \(k\gg 1\), \(f_k\) has exactly \(n\) zeros in the disc \(\left\{{{\left\lvert {z-z_0} \right\rvert} < \delta}\right\}\), counted with multiplicity. Moreover, these zeros converge to \(z_0\) as \(k\to \infty\).
The zeros of the sequence converge to the zeros of the limit.
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If \(\left\{{f_k}\right\}\) are univalent functions on \(\Omega\) converging normally to \(f\), then either
- \(f\) is univalent, or
- \(f\) is constant.