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Automorphic Carathéodory-Julia Theorem

Alexander Kheifets

Abstract

Let $w(ζ)$ be a function analytic on $\mathbb D$, $|w(ζ)|\le 1$. Let $|t_0|=1$. Assume that $w$ and $w'$ have nontangential boundary values $w_0$ and $w'_0$, respectively, at $t_0$, $|w_0|=1$. Then (Carathéodory - Julia) $t_0\dfrac{w'_0}{w_0}\ge 0$. The goal of this paper is to obtain a lower bound on this ratio if $w$ is character-automorphic with respect to a Fuchsian group (Theorem 6.1)

Automorphic Carathéodory-Julia Theorem

Abstract

Let be a function analytic on , . Let . Assume that and have nontangential boundary values and , respectively, at , . Then (Carathéodory - Julia) . The goal of this paper is to obtain a lower bound on this ratio if is character-automorphic with respect to a Fuchsian group (Theorem 6.1)
Paper Structure (8 sections, 26 theorems, 180 equations)

This paper contains 8 sections, 26 theorems, 180 equations.

Table of Contents

  1. Introduction.
  2. Pommerenke Type Theorems.
  3. Automorphic Properties of $m'_{t_0}$, $\varphi$ and $\Delta_{t_0}$.
  4. Existence of $\alpha$-automorphic Functions in $1+(t-t_0)H^2$.
  5. Kernels $k^\alpha_{t_0}$ and Their Properties.
  6. Automorphic Carathéodory-Julia: Necessary Condition.
  7. Additional Considerations.
  8. Appendix

Key Result

Theorem 1.2

Assume that group $\Gamma$ is of convergent type, that is the Blaschke product over the orbit of $\Gamma$ converges (This function is called the Green function of group $\Gamma$). Also assume that $\Gamma$ does not contain elliptic elements. Then for every unitary character $\alpha$ of $\Gamma$ there exists a non-zero $\alpha$-automorphic bounded analytic function if and only if$g'_{\zeta_0}(\ze

Theorems & Definitions (59)

  • Definition 1.1
  • Theorem 1.2: Widom-Pommerenke Widom71, Pom
  • Theorem 1.3: Kupin-Yuditskii
  • Theorem 1.4: Carathéodory - Julia
  • Definition 1.5
  • Theorem 1.6: Kh-Yud-2019
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 49 more