Abstract
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.
Citation
J. E. Pascoe. Meredith Sargent. Ryan Tully-Doyle. "A controlled tangential Julia–Carathéodory theory via averaged Julia quotients." Anal. PDE 14 (6) 1773 - 1795, 2021. https://doi.org/10.2140/apde.2021.14.1773
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