Related papers: Properties of the Ahlfors function
The classical Green's function associated to a simply connected domain in the complex plane is easily expressed in terms of a Riemann mapping function. The purpose of this paper is to express the Green's function of a finitely connected…
Let X be a Riemann surface of genus at most 1, i.e. X is the Riemann sphere or a torus. We construct a variety of examples of analytic functions g:W->X, where W is an arbitrary subdomain of X, that satisfy Epstein's "Ahlfors islands…
Here we advance the study of boundary the value problem for extremal functions of mean distortion and the associated Teichm\"uller spaces interpolating between the classical examples of extremal quasiconformal mappings, and the more recent…
We investigate the properties of a modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of an extremal function and state some of its properties. We obtain the integral formula…
In the general setting of a locally compact Abelian group $G$, the Delsarte extremal problem asks for the supremum of integrals over the collection of continuous positive definite functions $f: G \to \mathbb{R}$ satisfying $f(0) = 1$ and…
We study a problem of Jeong and Taniguchi asking to find all rational maps which are Ahlfors functions. We prove that the rational Ahlfors functions of degree two are characterized by having positive residues at their poles. We then show…
Using Ahlfors functions, Grunsky maps and the Bell representation theorem, we show that a certain subset of the rational maps of degree $n$ forms a trivial bundle over the moduli space of non-degenerate $n$-connected domains with one marked…
For each closed, positive (1,1)-current \omega on a complex manifold X and each \omega-upper semicontinuous function \phi on X we associate a disc functional and prove that its envelope is equal to the supremum of all…
Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus…
It is proved that the family of Ahlfors extremal mappings of a multiply connected region in the plane onto the unit disc can be expressed as a rational combination of two fixed Ahlfors mappings in much the same way that the family of…
The article introduces Ahlfors' generalization of the Schwarz lemma. With this powerful geometric tool of complex functions in one variable, we are able to prove some theorems concerning the size of images under holomorphic mappings,…
Ultrafunctions are a particular class of functions defined on a non-Archimedean field. They provide generalized solutions to functional equations which do not have any solutions among the real functions or the distributions. In this paper…
We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is…
We show how to express a conformal map of a general two connected domain in the plane such that neither boundary component is a point to a representative domain which has the virtue of having an explicit algebraic Bergman kernel function.…
We introduce extremal affine surface areas in a functional setting. We show their main properties. Among them are linear invariance, isoperimetric inequalities and monotonicity properties. We establish a new duality formula, which shows…
A multidimensional extremal problem in the idempotent algebra setting is considered which consists in minimizing a nonlinear functional defined on a finite-dimensional semimodule over an idempotent semifield. The problem integrates two…
A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Tur\'an extremal problems. A slight variation of the extremal problem is introduced and the…
In the paper "Bellman function for extremal problems in $\mathrm{BMO}$", the authors built the Bellman function for integral functionals on the $\mathrm{BMO}$ space. The present paper provides a development of the subject. We abandon the…
A holomorphic function f on a simply connected domain {\Omega} is said to possess a universal Taylor series about a point in {\Omega} if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta K outside…
We introduce Omega functions that generalize Euler Gamma functions and study the functional difference equation they satisfy. Under a natural exponential growth condition, the vector space of meromorphic solutions of the functional equation…