The Gehring-Hayman type theorems on complex domains

Pommerenke's theorem on Gromov hyperbolic domains

We establish a version of a classical theorem of Pommerenke, which is a diameter version of the Gehring-Hayman inequality on Gromov hyperbolic domains of $\mathbb{R}^n$. Two applications are given. Firstly, we generalize Ostrowski's Faltensatz to quasihyperbolic geodesics of Gromov hyperbolic domains. Secondly, we prove that unbounded uniform domains can be characterized in the terms of Gromov hyperbolicity and a naturally quasisymmetric correspondence on the boundary, where the Gromov boundary is equipped with a Hamenstädt metric (defined by using a Busemann function).

A note on the Hayman-Wu theorem

The Hayman-Wu theorem states that the preimage of a line or circle L under a conformal mapping from the unit disc to a simply-connected domain U has total Euclidean length bounded by an absolute constant. The best possible constant is known to lie in the interval [pi^2, 4 pi), thanks to work of Øyma and Rohde. Earlier, Brown Flinn showed that the total length is at most pi^2 in the special case in which U contains L. Let r be the anti-Möbius map that fixes L pointwise. In this note we extend the sharp bound pi^2 to the case where each connected component of the intersection of U with r(U) is b…

强Goldilocks域、定量可视性及其应用

Strongly Goldilocks Domains, quantitative Visibility, and Applications

受Goldilocks域相关研究启发，研究者定义强Goldilocks域子类，证明其定量可视性结果，推广Gehring-Hayman定理并给出Kobayashi距离下界估计。

Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains

If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension …

The distance ratio metric on some parts of $\mathbb C$ plane

We prove a generalization for some $\mathbb C$ domains of Gehring - Palka theorem on Moebius transformations regarding the distance ratio metric. Namely, we show that this theorem is valid for arbitrary holomorphic mappings $f : H \to H$ or $f : D \to D$ with the same Lipschitz constant $2$.

On the simultaneous Approximation of coefficients of schlicht functions

A modified Version of the Hardy-Littlewood tauberian Theorem is used to prove under which conditions the moduli of the coefficients |a(n)/n| of schlicht functions tend uniformly to their Hayman Indexes as n tends to infinity.

Weighted Bergman spaces and the $\bar{\partial}-$equation

We give a Hörmander type $L^2-$estimate for the $\bar{\partial}-$equation with respect to the measure $δ_Ω^{-α}dV$, $α<1$, on any bounded pseudoconvex domain with $C^2-$boundary. Several applications to the function theory of weighed Bergman spaces $A^2_α(Ω)$ are given, including a corona type theorem, a Gleason type theorem, together with a density theorem. We investigate in particular the boundary behavior of functions in $A^2_α(Ω)$ by proving an analogue of the Levi problem for $A^2_α(Ω)$ and giving an optimal Gehring type estimate for functions in $A^2_α(Ω)$. A vanishing theorem for $A^2_1…

On boundary behavior of generalized quasi-isometries

It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower $Q$-homeomorphisms $f$ between domains in $\overline{\Rn}=\Rn\cup\{\infty\}$, $n\geqslant2$, under integral constraints of the type $\intΦ(Q^{n-1}(x))\,dm(x)<\infty$ with a convex non-decreasing function $Φ:[0,\infty]\to[0,\infty]$. It is shown that integral conditions on the function $Φ$ found by us are not only sufficient but also necessary for a continuous extension of $f$ to the boundary. It is given also applications of the obtained results to the mappings with finite ar…

The Dirichlet problem for semi-linear equations

We study the Dirichlet problem for the semi--linear partial differential equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the complex plane $\mathbb C$ with continuous boundary data. We prove the existence of the weak solutions $u$ in the class $C\cap W^{1,2}_{\rm loc}(D)$ if a Jordan domain $D$ satisfies the quasihyperbolic boundary condition by Gehring--Martio. An example of such a domain that fails to satisfy the standard (A)--condition by Ladyzhenskaya--Ural'tseva and the known outer cone condition is given. We also extend our results to simply connected non-Jorda…

Some properties of analytic in a ball functions of bounded $\mathbf{L}$-index in joint variables

A concept of boundedness of the $\mathbf{L}$-index in joint variables (see in Bandura A. I., Bordulyak M. T., Skaskiv O. B. "Sufficient conditions of boundedness of L-index in joint variables", Mat. Stud. 45 (2016), 12--26. dx.doi.org/10.15330/ms.45.1.12-26) is generalized for analytic in a ball function. There are proved criteria of boundedness of the $\mathbf{L}$-index in joint variables which describe local behavior of partial derivatives and maximum modudus on a skeleton of a polydisc, properties of power series expansion. Also we obtained analog of Hayman's Theorem. As a result, they are …

斯梅尔中值猜想及其关于复多项式的对偶猜想

Smale's Mean Value Conjecture and its Dual Conjecture for Complex Polynomials

本文针对复多项式的斯梅尔中值猜想及其对偶猜想展开研究，通过改进Koebe型定理与马尔可夫型不等式，将对偶猜想的最优无条件下界从1/d²提升至(d−1/2)^(1/d)/d²。

拟共形性的度量定义与例外集

Metric definition of quasiconformality and exceptional sets

本文证明欧氏空间同胚拟共形的充要条件为每点存在收缩至该点的有界偏心率非中心开集列且像偏心率有界，推广Gehring定义，引入CNED集类并证明其为拟共形性例外集。