A general theory of osculation algorithms for conformal mapping

AbstractLet D be the family of simply connected regions D such that 0 ϵ D ⊂ E, the unit disk, and let D(ϱ) be the set of all D ϵ D such that dist(∂D, 0) = ϱ. An osculation family assigns to every D ϵ D a family F(D) of conformal maps h of D such that: (1) h(D) ϵ D, h(0) = 0, h′(0) > 0; (2) the inverse map h[−1] can be extended to an analytic function on E with values in E; (3) γ(ϱ): = infD ϵ D(ϱ)infh ϵ F(D) h′(0) satisfies infϱ ϵ Iγ(ϱ) > 1 for every I:=(0,β], where β < 1. (Example: the Koebe family, but there are many others.) We show that γ(ϱ) ⩽ (1 + ϱ)24ϱ for any osculation family, the bound being best possible for all ϱ ϵ (0,1]. An osculation algorithm constructs the Riemann map of a D ϵ D onto E as lim hn · hn − 1 ·…· h1, where D1:=D and Dn + 1: = hn(Dn),hn being any map in F(Dn). We prove convergence and, if Dn ϵ D(ϱn) and γ(ϱ) ⩾ 1 + α(1 − ϱ)2, show that 1 − ϱn ⩽ 8α−1ϱ1−2n−1, thereby generalizing and improving a result of Ostrowski (1929). If the unique solution φ = φ(ϱ) of φ = ϱ exp{[(1−γφ)/1+γφ)]Logγ}[γ:=γ(ϱ)] is increasing, we also show that ϱn ⩾ ϱ′n, where ϱ′1: = ϱ1, ϱ′n+1:=φ(ϱ′n), thus explaining the experimentally observed rapid convergence of the osculation algorithm during the first few iteration steps