Vol. 14, No. 6, 2021

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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
Abstract

Let f : 𝒟 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.

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