A controlled tangential Julia–Carathéodory theory via averaged Julia quotients

Pascoe, J. E.; Sargent, Meredith; Tully-Doyle, Ryan

doi:10.2140/apde.2021.14.1773

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A controlled tangential Julia–Carathéodory theory via averaged Julia quotients

J. E. Pascoe, Meredith Sargent and Ryan Tully-Doyle

Vol. 14 (2021), No. 6, 1773–1795

DOI: 10.2140/apde.2021.14.1773

Abstract

Let $f : 𝒟 \to Ω$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f (z)$ to the boundary of $Ω$ and the distance of $z$ to the boundary of $𝒟$ . A classical Julia–Carathéodory-type theorem states that if there is a sequence tending to $τ$ in the boundary of $𝒟$ along which the Julia quotient is bounded, then the function $f$ can be extended to $τ$ such that $f$ is nontangentially continuous and differentiable at $τ$ and $f (τ)$ is in the boundary of $Ω$ . We develop an extended theory when $𝒟$ and $Ω$ are taken to be the upper half-plane which corresponds to averaged boundedness of the Julia quotient on sets of controlled tangential approach, so-called $λ$ -Stolz regions, and higher-order regularity, including but not limited to higher-order differentiability, which we measure using $γ$ -regularity. Applications are given, including perturbation theory and moment problems.

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Keywords

tangential regularity, complex variables, Cauchy transforms

Mathematical Subject Classification 2010

Primary: 30E20, 47A10

Secondary: 47A55, 47A57

Milestones

Received: 18 February 2019

Revised: 6 February 2020

Accepted: 25 March 2020

Published: 7 September 2021

Authors

J. E. Pascoe
Department of Mathematics University of Florida Gainesville, FL United States

Meredith Sargent
Department of Mathematical Sciences University of Arkansas Fayetteville, AR United States

Ryan Tully-Doyle
Department of Mathematics and Physics University of New Haven West Haven, CT United States

Vol. 14, No. 6, 2021