#### Vol. 14, No. 6, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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
##### Abstract

Let $f:\mathsc{𝒟}\to \Omega$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f\left(z\right)$ to the boundary of $\Omega$ and the distance of $z$ to the boundary of $\mathsc{𝒟}$. A classical Julia–Carathéodory-type theorem states that if there is a sequence tending to $\tau$ in the boundary of $\mathsc{𝒟}$ 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 $f\left(\tau \right)$ is in the boundary of $\Omega$. We develop an extended theory when $\mathsc{𝒟}$ and $\Omega$ 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 $\lambda$-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.

##### Keywords
tangential regularity, complex variables, Cauchy transforms
##### Mathematical Subject Classification 2010
Primary: 30E20, 47A10
Secondary: 47A55, 47A57