The Julia-Wolff-Carathéodory theorem in polydisks

The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls inC ⁿ and by the author to holomorphic maps between strongly (pseudo)convex domains. Here we describe Julia-Wolff-Carathéodory theorems for holomorphic maps defined in a polydisk and with image either in the unit disk, or in another polydisk, or in a strongly convex domain. One of the main tools for the proof is a general version of the Lindelöf principle valid for not necessarily bounded holomorphic functions.     

A Class of Functions Holomorphic in the Disk

We introduce a new parametric representation of the class of holomorphic functions in the unit disk such that their Nevanlinna characteristic has a power growth near the boundary of the disk. The parameters of the obtained representation are determined explicitly by values of the function. In addition, the set of multipliers from the considered class to the Hardy and Bergman classes and the disk algebra is described completely. Bibliography: 10 titles.     

Boundary Distortion Estimates for Holomorphic Maps

We establish some estimates of the angular derivatives from below for holomorphic self-maps of the unit disk ${\mathbb {D}}$ at one and two fixed points of the unit circle provided there is no fixed point inside ${\mathbb {D}}$ . The results complement Cowen–Pommerenke and Anderson–Vasil’ev type estimates in the case of univalent functions. We use the method of extremal length and a semigroup approach to deriving inequalities for holomorphic self-maps of the disk which are not necessarily univalent using known inequalities for univalent functions. This approach allowed us to obtain a new Ossermans type estimate as well as inequalities for holomorphic self-maps which images do not separate the origin and the boundary.     

Lagrange Interpolation for the Disk Algebra: The Worst Case

We consider Lagrange interpolation polynomials for functions in the disk algebra with nodes on the boundary of the unit disk. In case that the closure of the set of nodes does not cover the boundary of the unit disk we prove that there exists a residual set of functions in the disk algebra, such that the Lagrange interpolation polynomials of each of these functions form a dense subset of the space of all holomorphic functions defined on the unit disk.     

Boundary behaviour of inner functions and holomorphic mappings

Let be a holomorphic function in the unit disk omitting a set of values of the complex plane. If has positive logarithmic capacity, R. Nevanlinna proved that has a radial limit at almost every point of the unit circle. If is any infinite set, we show that has a radial limit at every point of a set of Hausdorff dimension 1. A localization technique reduces this result to the following theorem on inner functions. If is an inner function omitting a set of values in the unit disk, then for any accumulation point of in the disk, there exists a set of Hausdorff dimension 1 of points in the circle where has radial limit . Received: 13 February 1997     

On a boundary analog of the Forelli theorem

A boundary analog of the Forelli theorem for real-analytic functions is established, i.e., it is demonstrated that each real-analytic function f defined on the boundary of a bounded strictly convex domain D in the multidimensional complex space with the one-dimensional holomorphic extension property along families of complex lines passing through a boundary point and intersecting D admits a holomorphic extension to D as a function of many complex variables.     

The Remainder Term of the Taylor Expansion for a Holomorphic Function Is Representable in Lagrange Form

We show that the remainder of the Taylor expansion for a holomorphic function can be written down in Lagrange form, provided that the argument of the function is sufficiently close to the interpolation point. Moreover, the value of the derivative in the remainder can be taken in the intersection of the disk whose diameter joins the interpolation point and the argument of the function and an arbitrary small angle whose bisectrix is the ray from the interpolation point through the argument of the function.     

Angular and unrestricted limits of one-parameter semigroups in the unit disk

We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier, under some additional condition (the position of the Denjoy–Wolff point) it was shown in [13] that elements of one-parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points. We prove stronger versions of these statements with no assumption on the position of the Denjoy–Wolff point. In contrast to many other problems, in the question of existence for unrestricted limits it appears to be more complicated to deal with the boundary Denjoy–Wolff point (the case not covered in [13]) than with all the other boundary fixed points of the semigroup.     

On the sharpness of Meier’s analogue of Fatou’s theorem

Meier’s topological analogue of Fatou’s theorem is shown to be sharp by exhibiting a bounded holomorphic function in the unit disk for which no point of a prescribed set of first category on the unit circle is a Meier point. Supported by the U. S. Army Research Office, Durham.     

Almost locally univalent functions

We study ALU holomorphic functions defined in the introduction, and prove two criteria for a function holomorphic in the unit disk to be ALU.     

A counterexample to the existence of peaking functions

We construct a smoothly bounded pseudoconvex domain whose boundary contains no complex analytic variety such that some boundary point admits no holomorphic peak function.

     

Some pathological examples of solutions to a Beltrami equation

We construct an example of a bounded solution to a uniformly elliptic Beltrami equation that has no nontangential limit values almost everywhere on the boundary of the unit disk and also an example of a solution to such an equation that is not identically zero and has zero nontangential limit values almost everywhere on the boundary of the unit disk. These examples show that, in the general case of the Hardy spaces of solutions to a uniformly elliptic Beltrami equation (and to more general noncanonical first-order elliptic systems), the usual statement of boundary value problems used for holomorphic and generalized analytic functions is ill-posed.     

A theorem in fine potential theory and applications to finely holomorphic functions

Consider the map from the fine interior of a compact set to the measures on the fine boundary given by Balayage of the unit point mass onto the fine boundary (the Keldych measure). It is shown that for any point in the domain there is a compact fine neighborhood of the point on which the map is continuous from the initial topology on the compact set to the norm topology on measures. In this paper we only prove a rather special case, the method could easily be generalized to more abstract potential spaces. One consequence of this result is a Hartog-type theorem for finely harmonic functions. We use the Hartog theorem, rational approximation theory, and results proved in a previous paper by the author to prove that the derivative of a finely holomorphic function exists everywhere and is finely holomorphic.     

Backward iteration in strongly convex domains

We prove that a backward orbit with bounded Kobayashi step for a hyperbolic, parabolic or strongly elliptic holomorphic self-map of a bounded strongly convex C² domain in Cd necessarily converges to a repelling or parabolic boundary fixed point, generalizing previous results obtained by Poggi-Corradini in the unit disk and by Ostapyuk in the unit ball of Cd.     

Yet another proof for a theorem of Landau on Koebe domains

The Koebe domain of a family of functions, holomorphic on the unit disk, is the largest domain that is contained in the image of the unit disk for every function of the family. In this note, we furnish a geometric proof of a classical theorem due to Landau on the Koebe domains for certain families of holomorphic functions. The method of proof involves our recently obtained results concerning estimates for hyperbolic metrics on subdomains.     

Bolzano's Theorems for Holomorphic Mappings(To Ham Brezis,with friendship and admiration)

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions. A more complicated proof, using Cauchy's argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from C~n to C~n are obtained using Brouwer degree.     

Bolzano's Theorems for Holomorphic Mappings

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions.A more complicated proof,using Cauchy's argument principle,provides uniqueness of the zero,when the sign conditions on the boundary are strict.Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions.Extensions to holomorphic mappings from Cn to Cn are obtained using Brouwer degree.     

Holomorphic extension from the sphere to the ball

Real analytic functions on the boundary of the sphere which have separate holomorphic extension along the complex lines through a boundary point have holomorphic extension to the ball. This was proved in Baracco (2009) [4] by an argument of CR geometry. We give here an elementary proof based on the expansion in holomorphic and antiholomorphic powers.     

On the hyperbolic order

We define the hyperbolic order of any locally injective holomorphic function between arbitrary hyperbolic domains of the complex plane and study the relation between the hyperbolic order and the Schwarzian derivative for locally injective holomorphic functions from the unit disk into itself.     

A sharp Schwarz inequality on the boundary

A number of classical results reflect the fact that if a holomorphic function maps the unit disk into itself, taking the origin into the origin, and if some boundary point maps to the boundary, then the map is a magnification at . We prove a sharp quantitative version of this result which also sharpens a classical result of Loewner.