Digons and angular derivatives of analytic self-maps of the unit disk

Lower Schwarz-Pick estimates and angular derivatives

The well-known Schwarz-Pick lemma states that any analytic mapping $ϕ$ of the unit disk $U$ into itself satisfies the inequality $$|ϕ'(z)|\leq \frac{1-|ϕ(z)|^2}{1-|z|^2}, \quad z\in U.$$ This estimate remains the same if we restrict ourselves to univalent mappings. The lower estimate is $|ϕ'(z)|\geq 0$ generally or $|ϕ'(z)|> 0$ for univalent functions. To make the lower estimate non-trivial we consider univalent functions and fix the angular limit and the angular derivative at some points of the unit circle. In order to obtain sharp estimates we make use of the reduced modulus of a digon.

单位圆盘上一参数半群花瓣的角导数问题

The angular derivative problem for petals of one-parameter semigroups in the unit disk

本文研究了单位圆盘全纯自映射的一参数半群花瓣的角导数问题，分别基于内蕴双曲几何与后向动力学、Koenigs 函数渐近行为刻画了双曲与抛物花瓣共形性的充要条件。

Angular Derivatives and Boundary Values of H(b) Spaces of Unit Ball of $\mathbb{C}^n$

In this work we study deBranges-Rovnyak spaces, $H(b)$, on the unit ball of $\mathbb{C}^n$. We give an integral representation of the functions in $H(b)$ through the Clark measure on $S^n$ associated with $b$. A characterization of admissible boundary limits is given in relation with finite angular derivatives. Lastly, we examine the interplay between Clark measures and angular derivatives showing that Clark measure associated with $b$ has an atom at a boundary point if and only if $b$ has finite angular derivative at the same point.

内函数、莫比乌斯畸变与角导数

Inner Functions, Möbius Distortion and Angular Derivatives

本文证明内函数有限$\mathcal{L}(p)$熵与累积莫比乌斯畸变属于$L^p$等价，并研究了使对应奇异内函数有有限$\mathcal{L}(p)$熵的正奇异测度支撑。

角导数密度的尖锐熵条件

A Sharp Entropy Condition For The Density Of Angular Derivatives

本文证明单位圆盘全纯自映射的有限 Carathéodory 角导数点集在熵可积条件下必为可数个有限熵 Beurling-Carleson 集之并，并构造反例说明该刻画是精确的。

Matrix-valued Aleksandrov--Clark measures and Carathéodory angular derivatives

This paper deals with families of matrix-valued Aleksandrov--Clark measures $\{\boldsymbolμ^α\}_{α\in\mathcal{U}(n)}$, corresponding to purely contractive $n\times n$ matrix functions $b$ on the unit disc of the complex plane. We do not make other apriori assumptions on $b$. In particular, $b$ may be non-inner and/or non-extreme. The study of such families is mainly motivated from applications to unitary finite rank perturbation theory. A description of the absolutely continuous parts of $\boldsymbolμ^α$ is a rather straightforward generalization of the well-known results for the scalar case (…