复变函数

We study Chebyshev constants and transfinite diameter on the graph of a polynomial mapping $f\colon\mathbb{C}^2\to\mathbb{C}^2$. We show that two transfinite diameters of a compact subset of the graph (i.e., defined with respect to two…

We present an explicit method to compute the (Siciak-Zaharjuta) extremal function of a real convex polytope in terms of supporting simplices and strips. We use this to give a new proof of the existence of extremal ellipses associated to the…

We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial degree. We also generalize Leja points to this setting. As an application of our main formula,…

Recently, we introduced domains of slice regularity in the space $\mathbb{H}$ of quaternions and also proved that domains of slice regularity satisfy a symmetry with respect to paths, called $2$-path-symmetry. In this paper, we give a full…

We give a description of the intersection of the zero set with the unit sphere of a zero-free polynomial in the unit ball of $\mathbb{C}^n$. This description leads to the formulation of a conjecture regarding the characterization of…

A new approach is given to property $(P_q)$ defined by Catlin for $q=1$ in a global and by Sibony in a local context, subsequently extended by Fu-Straube for $q>1$. This property is known to imply compactness and global regularity in the…

We discuss the zero sets of two-variable polynomials as they relate to an approximation problem in the Hardy space on the bidisk.

Let $f$ and $g$ be analytic functions on the open unit disc $\mathbb D$ such that $|f|=|g|$ on a set $A$. We give an alternative proof of the result of Perez that there exists $c$ in the unit circle $\mathbb T$ such that $f=cg$ when $A$ is…

In this article, we give some necessary conditions for the concavity property of minimal $L^2$ integrals degenerating to partial linearity, a charaterization for the concavity degenerating to partial linearity for open Riemann surfaces, and…

Let $\alpha$ \in (0; 1). We show that any $\alpha$-H\"older homeomorphism from the unit circle in the plane to the plane can be extended to an $\alpha$-H\"{o}lder homeomorphism from the whole unit disc.

Let $\Omega$ be a region in the complex plane $\mathbb C$ and let $\{\Phi_t \}_{t\ge 0}$ be a continuous semigroup of functions on $\Omega$; that is, $\Phi_t:\Omega\to\Omega$ is holomorphic for every $t\ge 0$, $\Phi_0(z)=z$, for every…

In this article, we first establish a generalized Bohr inequality and examine its sharpness for a class of analytic functions $f$ in a simply connected domain $\Omega_\gamma,$ where $0\leq \gamma<1$ with a sequence $\{\varphi_n(r)…

Arakeljan's Theorem provides conditions on a relatively closed subset $F$ of a domain $G\subset\mathbb{C}$, such that any continuous function $f:F\rightarrow\mathbb{C}$ that is analytic in $F^\circ$, can be approximated by analytic…

Inspired by a result by Sz\H{o}ke, we give potential-theoretic characterizations of the dimension of the Bergman space of holomorphic sections of a restriction of a holomorphic line bundle of $\mathbb{P}^1$ to some open set…

We set a framework for the study of Hardy spaces inherited by complements of analytic hypersurfaces in domains with a prior Hardy space structure. The inherited structure is a filtration, various aspects of which are studied in specific…

For the real part of the Cauchy-type integral that is known to be the logarithmic potential of the double layer, a necessary and sufficient condition for the continuous extension to the Ahlfors-regular boundary is established.

In this paper, we prove that a spirallike circularlike domain is Kobayashi hyperbolic if and only if its core is empty. In particular, we show that such a domain is Kobayashi hyperbolic if and only if it is (biholomorphic to) a bounded…

On a K\"ahler manifold we consider the problems of maximizing/minimizing Monge--Amp\`ere energy over certain subsets of the space of K\"ahler potentials. Under suitable assumptions we prove that solutions to these variational problems…

Sobolev mappings exhibiting only pointwise quasiregularity-type bounds have arisen in various applications, leading to a recently developed theory of quasiregular values. In this article, we show that by using rescaling, one obtains a…

In this paper, we show that there exist transcendental meromorphic functions with a cycle of 2-periodic Fatou components, where one is simply connected while the other is doubly connected. In particular, the doubly connected Fatou component…