Exceptional sets for the derivatives of Blaschke products
Emmanuel Fricain, Javad Mashreghi
我们得到了Blaschke乘积的对数导数$B'(z)/B(z)$的增长估计为$|z| \to 1$和$z$避免了一些例外集.
We obtain growth estimates for the logarithmic derivative $B'(z)/B(z)$ of a Blaschke product as $|z| \to 1$ and $z$ avoids some exceptional sets.
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A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke products: those that maximize the difference between the maximum and minimum of the derivative, and those that minimize it. Both classes turn out to have the same algebraic structure, being the quotient of two hypergeometric polynomials.
The relationship between the distribution of zeros of an infinite Blaschke product $B$ and the inclusion in weighted Bergman spaces $A_α^p$ of the derivative of $B$ or the derivative of functions in its model space $H^2 \ominus BH^2$ is investigated.
On critical points of Blaschke products
得到了零点位于Stolz型区域内的Blaschke积的导数的一个上界.我们表明，衍生物属于空间的解析函数的单位磁盘，最近推出的\cite{FG}。作为结果，我们得到了这类Blaschke乘积的临界点的Blaschke型条件.
Blaschke products with derivative in function spaces
设$B$是具有零的Blaschke积$\{a_n\}$。如果$B' \in A^p_α$对于某些$p$和$α$，则表明$\sum_n (1 - |a_n|)^β < \infty$对于$β$的适当值。此外，如果$\{a_n\}$是均匀离散的，如果$B' \in H^p$或$B' \in A^{1+p}$对于任何$p \in (0,1)$，它表明，$\sum_n (1 - |a_n|)^{1-p} < \infty$。
Integral means of the derivatives of Blaschke products
本文研究了一类Blaschke乘积的导数的积分平均的增长率，推广了几个经典结果。此外，我们还得到了由Blaschke积$B$生成的模型子空间$K_B$中函数导数的积分平均的增长率
We establish a sharp upper bound for the absolute value of the derivative of the finite Blaschke product, provided that the critical values of this product lie in a given disk.
Maximal Blaschke Products
考虑了经典的单位圆盘上所有有界解析函数的集合在一个不动点处的导数最大化问题，该集合具有指定的临界点。我们证明了极值函数本质上是唯一的，并且总是不可破坏的Blaschke乘积。这一结果推广了Nehari-Schwarz引理，并得到了一类新的Blaschke积，称为极大Blaschke积.我们建立了极大Blaschke积的一些性质，这些性质表明极大Blaschke积是有限Blaschke积类的一个适当的无限推广.
A finite Blaschke product is a product of finitely many automorphisms of the unit disk. This brief survey covers some of the main topics in the area, including characterizations of finite Blaschke products, approximation theorems, derivatives and residues of finite Blaschke products, geometric localization of zeros, and selected other topics.
On Blaschke products with derivatives in Bergman spaces with normal weights
我们推广了一个著名的充分条件插值序列的Hilbert Bergman空间到其他Bergman空间与正常的重量（定义由Shields和Williams），并获得了新的结果，关于衍生物的Blashke产品或一般的内部功能在这样的空间。我们还应用对偶技巧得到了这类结果，并得到了关于插值Blaschke积的新结果。
Let $\mathscr J$ be the set of inner functions whose derivatives lie in Nevanlinna class. In this note, we show that the natural map $F \to \text{Inn}(F'): \mathscr J/\text{Aut}(\mathbb{D}) \to \text{Inn}/S^1$ is is injective but not surjective. More precisely, we show that that the image consists of all inner functions of the form $BS_μ$ where $B$ is a Blaschke product and $S_μ$ is the singular factor associated to a measure $μ$ whose support is contained in a countable union of Beurling-Carleson sets. Our proof is based on extending the work of D. Kraus and O. Roth on maximal Blaschke produc…
Geometric subfamily of locally univalent functions, Blaschke products and quasidisk
本文研究了局部单值函数的几何子族F(α)，证明其与Blaschke乘积和Schwarzian导数的联系，并给出了准圆盘和预Schwarzian导数的精确界。
Geometric subfamily of functions convex in some direction and Blaschke products
本文研究满足特定规范化与曲率条件的局部单叶解析函数族，证明其具有涉及 Blaschke 乘积、Schwarz 导数及单叶调和映射的若干优雅性质。
We obtain estimates for integrals of derivatives of rational functions in multiply connected domains in the plane. A sharp order of the growth is found for the integral of the modulus of the derivative of a finite Blaschke product in the unit disk. We also extend the results of E.P. Dolzhenko about the integrals of the derivatives of rational functions to a wider class of domains, namely, to domains bounded by rectifiable curves without zero interior angles, and show the sharpness of the obtained results.
Extremal solutions of Nevalinna-Pick problems and certain classes of inner functions
考虑尺度Nevanlinna-Pick插值问题，设$Π$为Blaschke积，其零点为问题的节点。证明了如果$Π$属于某类内函数，则问题的极值解或大多数极值解属于同一类。三个不同的经典类被认为是：内函数，其衍生物是在一定的Hardy空间，指数Blaschke产品，也是众所周知的类$α$ -Blaschke产品，为$0<α<1$。
For entire Dirichlet series of the form $F(z)=\sum\limits_{n=0}^{+\infty} a_{n}e^{zλ_n},\ 0\leλ_n\uparrow+\infty\ (n\to+\infty)$, we establish conditions under which the relation $$ F(x+iy)=(1+o(1))a_{ν(x,F)}e^{(x+iy)λ_{ν(x,F)}} $$ is true as $x\to+\infty$ outside some set $E$ such that $\text{ h-meas }(E)=\int_{E}dh(x)<+\infty$ uniformly in $y\in\Bbb{R}$, where $h(x)$ is positive continuous function increasing to $+\infty$ on $[0,+\infty)$ with non-decreasing to $+\infty$ derivative.
Vanishing, Unbounded and Angular Shifts on the Quotient of the Difference and the Derivative of a Meromorphic Function
本文研究亚纯函数差分与导数之商的邻近函数，证明在周期消失或趋于无穷时其趋于零或为小函数，并给出角向偏移估计及亏量关系等应用。
Estimates on the Growth of Meromorphic Solutions of Linear Differential Equations with density conditions
本文给出了Hayman和作者在有限圆盘上或开平面上得到的系数为亚纯的齐次线性微分方程亚纯解的逐点估计的一种可供选择的简单方法.特别是，我们的估计一般给出更好的上界高阶导数的亚纯解的考虑，是有效的，然而，一个例外的有限对数密度集之外。估计再次表明，在无穷远点有正亏的亚纯解的增长性可以根据解在原点或原点附近的初始条件和系数的特征函数来估计。
The well-known $\mathcal{E}$-sets introduced by Hayman in 1960 are collections of Euclidean discs in the complex plane with the following property: The set of angles $θ$ for which the ray $\arg(z)=θ$ meets infinitely many discs in a given $\mathcal{E}$-set has linear measure zero. An important special case of an $\mathcal{E}$-set is known as the $R$-set. These sets appear in numerous papers in the theories of complex differential and functional equations. This paper offers a continuous transition from $\mathcal{E}$-sets to $R$-sets, and then to much thinner sets. In addition to rays, plane cur…
Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure
本文肯定回答了Grosse-Erdmann关于单位圆盘内Wiman-Valiron不等式的两个问题，在有限对数测度例外集外建立迭代对数不等式，并给出Rosenbloom复合估计的圆盘类比，关键利用Khinchin族方差界，且指数1/2不可改进。
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