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A controlled
tangential Julia–Carathéodory theory via averaged Julia
quotients
J. E. Pascoe, Meredith Sargent and Ryan Tully-Doyle |
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Vol. 14 (2021), No. 6, 1773–1795
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Abstract |
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Let be a complex analytic function. The Julia quotient is given by the ratio between the distance of to the boundary of and the distance of to the boundary of . A classical Julia–Carathéodory-type theorem states that if there is a sequence tending to in the boundary of along which the Julia quotient is bounded, then the function can be extended to such that is nontangentially continuous and differentiable at and is in the boundary of . We develop an extended theory when and 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 -Stolz regions, and higher-order regularity, including but not limited to higher-order differentiability, which we measure using -regularity. Applications are given, including perturbation theory and moment problems. |
Keywords
tangential regularity, complex variables, Cauchy transforms
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Mathematical Subject Classification 2010
Primary: 30E20, 47A10
Secondary: 47A55, 47A57
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Milestones
Received: 18 February 2019
Revised: 6 February 2020
Accepted: 25 March 2020
Published: 7 September 2021
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Authors |
| J. E. Pascoe | |
| Department of Mathematics University of Florida Gainesville, FL United States |
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| Meredith Sargent | |
| Department of Mathematical
Sciences University of Arkansas Fayetteville, AR United States |
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| Ryan Tully-Doyle | |
| Department of Mathematics and
Physics University of New Haven West Haven, CT United States |
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