The Poincaré Metric of Plane Domains

Univalence criteria in multiply-connected domains

“…Once again this is independent of the choice of p. The hyperbolic norm of an analytic function in D is defined just as in (4). Our main result is as follows.…”

“…Lower bounds for pD in terms of the distance to dD in the case when D is multiply-connected have been recently obtained in [4]. Once again the quasihyperbolic norm is monotonie;…”

On the Uniform Perfectness of the Boundary of Multiply Connected Wandering Domains

“…The concept of uniform perfectness was introduced by Beardon and Pommerenke [6,15] and has found many applications in complex analysis. It was proved by Mañé and da Rocha [13], Hinkkanen [11] and Eremenko [10], with different proofs, that Julia sets of rational functions are uniformly perfect.…”

The limit sets of Schottky quasiconformal groups are uniformly perfect

“…n , the domain 1 , ∞], where s > 0. We use τ n (s) to denote the modulus of the family of all curves connecting the two complementary components of D, and we also briefly write τ (s) instead of τ n (s).…”

“…A closed subset A of the complex plane C is said to be uniformly perfect if there exists a constant c > 0 such that A ∩ {z : cr ≤ |z − a| ≤ r} = ∅ , for any a ∈ A and 0 < r < diam (A), where diam (A) is the Euclidean diameter of the set A. Pommerenke proved in [10] . The idea of uniform perfectness was first introduced in [1]. In that paper, Beardon and Pommerenke showed that the boundary of a domain Ω ⊂ C not containing a neighborhood of infinity is uniformly perfect if, and only if, there exists a positive number c such that…”