Abstract
We show that the orthogonal speed of semigroups of holomorphic selfmaps of the unit disc is asymptotically monotone in most cases. Such a theorem allows to generalize previous results of D. Betsakos and D. Betsakos, M. D. Contreras and S. Díaz-Madrigal and to obtain new estimates for the rate of convergence of orbits of semigroups.
Highlights
The point 0 can be replaced with any z ∈ D
If α, β > 0, h(D) = Wα,β is quasi-symmetric with respect to vertical axes and the result can be obtained from Theorem 1.2 (and the explicit computation of the orthogonal speed of)
The estimate (2) in Theorem 1.2 is not a relation between the orthogonal speeds of and (but between the orthogonal speeds of and the total speed of), and can be obtained by the methods illustrated in this paper
Summary
We briefly recall the basics of the theory of semigroups of holomorphic selfmaps of the unit disc, as needed for our aims. A semigroup (φt) without fixed points in D is called non-elliptic. If (φt) is a non-elliptic semigroup, φt has the same Denjoy-Wolff point τ ∈ ∂ D, for all t > 0. Let (φt) be a non-elliptic semigroup in D. Up to conjugate with a rotation, we can assume that the Denjoy-Wolff point of (φt) is 1. This is, the Denjoy-Wolff Theorem version in H (see [10, Thm. 1.7.8]). If (φt) is a non-elliptic semigroup in D, there exists a (essentially unique) univalent function h : D → C such that (1) h(φt(z)) = h(z) + it for all z ∈ D, t ≥ 0, (2) t≥0(h(D) − it) = Ω, where Ω is either a vertical strip, or a vertical half-plane or C.
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