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Preprint
Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carath\'eodory theory
24 Sep 2018
Abstract
Let $f: D \rightarrow \Omega$ be a complex analytic function. The Julia
quotient is given by the ratio between the distance of $f(z)$ to the boundary
of $\Omega$ and the distance of $z$ to the boundary of $D.$ A classical
Julia-Carath\'eodory type theorem states that if there is a sequence tending to
$\tau$ in the boundary of $D$ along which the Julia quotient is bounded, then
the function $f$ can be extended to $\tau$ such that $f$ is nontangentially
continuous and differentiable at $\tau$ and $f(\tau)$ is in the boundary of
$\Omega.$ We develop an extended theory when $D$ and $\Omega$ are taken to be
the upper half plane which corresponds to amortized boundedness of the Julia
quotient on sets of controlled tangential approach, so-called $\lambda$-Stolz
regions, and higher order regularity, including but not limited to higher order
differentiability, which we measure using $\gamma$-regularity. Applications are
given, including perturbation theory and moment problems.
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Details
- Title
- Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carath\'eodory theory
- Creators
- J. E PascoeMeredith SargentRyan Tully-Doyle
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021879755604721