The branched Schwarz lemma: A classical result via circle packing
The seminal paper concerning the connection with circle packings of domains in the plane and analytic functions is that of \textit{B. Rodin} and \textit{D. Sullivan} [J. Differ. Geom. 26, 349-360 (1987; Zbl 0694.30006)] in which they proved a conjecture of W. Thurston on the approximation of conformal mappings via circle packings. Here, the authors develop aspects of a ``discrete analytic function theory'' based on circle packings to parallel classical function theory. In particular, they develop analogues in terms of circle packings of the following classical result (see, for example, \textit{Z. Nehari} [Duke Math. J. 14, 1035-1049 (1947; Zbl 0029.12302)]). (1) The branched Schwarz's lemma: Let \(f\), \(b\) be analytic functions mapping the unit disc \({\mathbf D}\) into itself, where \(f(0)=b(0)=0\), \(b\) is a finite Blaschke product, and the branch points of \(b\) form a subset (taking account of multplicity) of those of \(f\). Then \(|f'(0)|\leq|b'(0)|\); if \(|b'(0)|\neq 0\), equality holds iff \(f=\lambda b\) for some unimodular constant \(\lambda\). (2) The branched distortion lemma: Let \(f\) be an analytic function in \({\mathbf D}\) with \(f(0)=0\), \(r>0\), \(\Omega\) the component of \(f^1(r{\mathbf D})\) that contains \(0\), and \(f|_\Omega:\Omega\to r{\mathbf D}\) a proper mapping. Let \(b\) be a finite Blaschke product whose branch points consist of the branch points of \(f\) that lie in \(\Omega\) (taking account of multiplicity), and with \(b(0)=0\). Then \(|f'(0)|\geq r|b'(0)|\); if \(|b'(0)|\neq 0\), equality holds iff \(f=(\lambda r)b\) for some unimodular constant \(\lambda\). The authors assert that the discrete theory inspires and helps to prove classical results, and look forward to new ``classical'' results turning up in this way. First the authors review the definitions and notation associated with circle packings. They then formulate and prove discrete versions of the standard Schwarz and distortion lemmas, along with a number of related results on circle packings: one relates to the approximation of derivatives of analytic functions by `ratio functions', the other establishes the uniqueness of certain extremal hyperbolic circle packings. They then review the notion of branched circle packings and their use in the approximation of analytic functions. They state and prove the discrete versions of results (1) and (2) above, and then obtain (1) and (2) from the discrete versions by approximation. Finally, they point out that it would be valuable to obtain more quantitative information about the discrete inequalities involved in the arguments.
scientific article; zbMATH DE number 838727