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On the rate of convergence of parabolic semigroups of holomorphic functions

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Abstract

Let \(\{\phi _t\}_{t\ge 0}\) be a semigroup of holomorphic self-maps of the unit disk \({\mathbb D}\) with Denjoy–Wolff point 1 and associated planar domain \(\Omega \). We further assume that for all \(t>0\), \(\phi _t^\prime (1)=1\) (angular derivative), namely the semigroup is parabolic. We study the rate of convergence of the semigroup to 1. We prove then that for every \(z\in {\mathbb D}\), there exists a positive constant \(C\) such that \(|\phi _t(z)-1|\le C\;t^{-1/2}\). If, in addition, \(\Omega \) is contained in a half-plane, then we prove a stronger estimate: \(|\phi _t(z)-1|\le C\;t^{-1}\).

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Correspondence to Dimitrios Betsakos.

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Betsakos, D. On the rate of convergence of parabolic semigroups of holomorphic functions. Anal.Math.Phys. 5, 207–216 (2015). https://doi.org/10.1007/s13324-014-0095-8

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  • DOI: https://doi.org/10.1007/s13324-014-0095-8

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