Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carath\'eodory theory

TL;DR

This paper extends Julia-Carathéodory theory to include controlled tangential approaches and higher order regularity in the upper half-plane, with applications in perturbation theory and moment problems.

Contribution

It introduces a new framework for analyzing boundary behavior of analytic functions using $eta$-Stolz regions and $eta$-regularity, expanding classical results.

Findings

Extended Julia-Carathéodory theorem for $eta$-Stolz regions.

Higher order regularity results including differentiability.

Applications to perturbation theory and moment problems.

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.