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

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.

tangential regularity, complex variables, Cauchy transforms
Mathematical Subject Classification 2010
Primary: 30E20, 47A10
Secondary: 47A55, 47A57
Received: 18 February 2019
Revised: 6 February 2020
Accepted: 25 March 2020
Published: 7 September 2021
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