On this website I want to bring together a variety of my results. I introduced the notion of normal function of several complex variables in the articles published in 1981. According to Professor Gavrilov V. I.​ (PDF) those articles should be considered as the beginning of the theory of normal functions of several complex variables ‐ AMS Subject Classification 32A18.

Marty’s criterion in $\mathbb{C}^{n}$ is one of the main components to all proofs. Applications of Marty’s criterion allowed me to generalize some of the classical theorems to the case of several complex variables, including Zalcman’s lemma.

Logical map of my key results

Bibliography

A Canonical Characterization of Normal Functions

(joint work with S.G.Krantz)
arXiv (2026) DOI

We characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. We then generalize this result to a characterization of normal functions according to behavior on analytic discs. A simple proof of an old theorem of Hartog’s that a formal power series at 0 in $\mathbb{C}^n$ is convergent if its restriction to each complex line through the origin is convergent are given.

Book “A Second Course in Complex Analysis”

by Peter V.​ Dovbush (Author), Steven G.​ Krantz (Author)
Book will be published: 27 April 2026 by Chapman & Hall (Order now)

Few other books purport to be a second course in complex analysis. This book differs in that it covers more modern topics and is more geometric in focus. Most texts on complex variable theory contain the same material. However, complex analysis is a vast and diverse subject with a long history and many aspects. A second course will benefit students and introduce these new topics that they might not otherwise experience.

Lars Ahlfors alone invented many new parts of the subject; Lipman Bers made decisive contributions, and there are many others. It is easy to justify a “second course” in complex analysis. That is what this book purports to be.

Some of the topics presented here are:

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  harmonic measure

• •

  extremal length

• •

  Riemann surfaces

• •

  uniformization

• •

  automorphism groups

• •

  the Schwarz lemma and its generalizations

• •

  analytic capacity

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  the Bergman theory

• •

  invariant metrics

• •

  Picard’s theorem

• •

  the boundary Schwarz lemma

The goal is to expose the reader to unfamiliar parts of the subject of complex variables and perhaps to pique interest in further reading. As with the authors’ other books, not only theorems and proofs are included, but also many examples and some exercises. Numerous graphics illustrate the key ideas.

Book “One Complex Variable from the Several Variable Point of View”

by Peter V.​ Dovbush (Author), Steven G.​ Krantz (Author)
Published: 29 June 2025 by Chapman & Hall (Order now)

Traditionally speaking, those who study the function theory of one complex variable spend little or no time thinking about several complex variables. Conversely, experts in the function theory of several complex variables do not consider one complex variable. One complex variable is the inspiration and testing ground for several complex variables, and several complex variables is the natural generalization of one complex variable.

The authors’ thesis here is these two subject areas have much in common. They can gain a lot by learning to communicate. These two fields are logically connected, and each can be used to explain and to put the other into context. This is the purpose of this book.

The point of view and the methodology of the two subject areas are quite different. One complex variable is an aspect of traditional hard analysis. Several complex variables is more like algebraic geometry and differential equations, with some differential geometry thrown in. The authors intend to create a marriage of the function theory of one complex variable and the function theory of several complex variables, leading to a new and productive dialogue between the two disciplines.

The hope is this book to foster and develop this miscegenation in a manner that leads to new collaborations and developments. There is much fertile ground here, and this book aims to breathe new life into it.

The central role of Montel’s theorem in complex function theory

(joint work with S.G.Krantz)
Complex Var. Elliptic Equ. 70(9), 1615–1625. DOI

We begin with some new characterizations of normal families. Then we shift gears to see that normal families lie at the basis of complex function theory. Whereas normal families are usually treated as an ancillary topic in complex function theory, we show here that many of the main ideas in the subject may be derived from Montel’s theorem. Several consequences are derived.

Book “The Geometric Theory of Complex Variables”

by Peter V.​ Dovbush (Author), Steven G.​ Krantz (Author)
Published: 29 January 2025 by Springer Cham (Order now)

This book provides the reader with a broad introduction to the geometric methodology in complex analysis. It covers both single and several complex variables, creating a dialogue between the two viewpoints.

Regarded as one of the ’grand old ladies’ of modern mathematics, complex analysis traces its roots back 500 years. The subject began to flourish with Carl Friedrich Gauss’s thesis around 1800. The geometric aspects of the theory can be traced back to the Riemann mapping theorem around 1850, with a significant milestone achieved in 1938 with Lars Ahlfors’s geometrization of complex analysis. These ideas inspired many other mathematicians to adopt this perspective, leading to the proliferation of geometric theory of complex variables in various directions, including Riemann surfaces, Teichmüller theory, complex manifolds, extremal problems, and many others.

This book explores all these areas, with classical geometric function theory as its main focus. Its accessible and gentle approach makes it suitable for advanced undergraduate and graduate students seeking to understand the connections among topics usually scattered across numerous textbooks, as well as experienced mathematicians with an interest in this rich field.

Construction of the Riemann Map

(joint work with S.G.Krantz)
Complex Anal.​ Oper.​ Theory 18, 165(2024).​ DOI

We do not give the proof of famous Riemann mappings theorem, for every simply connected region in $\mathbb{C}$ but we give a series of increasingly better approximations of the Riemann map in the Riemann mapping theorem. The idea is to approximate a domain by a polygon using a Whitney decomposition, then use the Schwarz—Christoffel formula to compute the map onto the polygonal approximation, and finally use the Carathéodory convergence theorem to obtain the Riemann map in the limit. The suggested method will have easy and effective applications in practice.

Normal Holomorphic Mappings in Complex Space

(joint work with S.G.Krantz)
Complex Anal. Oper. Theory 18, 28 (2024). DOI

We study normal holomorphic mappings on complex spaces and complex manifolds. Applications are provided.

Book “Normal Families and Normal Functions”

by Peter V.​ Dovbush (Author), Steven G.​ Krantz (Author)
Published: February 27, 2024 by Chapman & Hall (Order now)

This book centers on normal families of holomorphic and meromorphic functions and also normal functions. The authors treat one complex variable, several complex variables, and infinitely many complex variables (i.e., Hilbert space).

The theory of normal families is more than 100 years old. It has played a seminal role in the function theory of complex variables. It was used in the first rigorous proof of the Riemann mapping theorem. It is used to study automorphism groups of domains, geometric analysis, and partial differential equations.

The theory of normal families led to the idea, in 1957, of normal functions as developed by Lehto and Virtanen. This is the natural class of functions for treating the Lindelof principle. The latter is a key idea in the boundary behavior of holomorphic functions.

This book treats normal families, normal functions, the Lindelof principle, and other related ideas. Both the analytic and the geometric approaches to the subject area are offered. The authors include many incisive examples.

The book could be used as the text for a graduate research seminar. It would also be useful reading for established researchers and for budding complex analysts.

Normal families and normal functions in $\mathbb{C}^{n}$

(joint work with S.G.Krantz)
The Tenth Congress of Romanian Mathematicians, June 30 ‐ July 5, 2023, Piteşti, Romania PDF

This talk surveys various generalizations and strengthenings of the classical theorems of Marty, Zalcman, Zalcman‐Pang, Montel, Schottky, Schwick, Royden, Mandelbrojt, Lohwater‐Pommerenke, Lehto‐Virtanen and Lindelöf with an emphasis on some surprising recent developments which are contained in my papers:

• •

  Complex Var. Elliptic Equ., 65.5 (2020), 66.12 (2021), 67.1 (2022),

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  J. Geom. Anal. 31.5 (2021),

• •

  Rev. Roumaine Math. Pures Appl. 67 (2022), no. 1‐2,

• •

  Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1981),

• •

  Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1 (1981).

Generalization of Lohwater‐Pommerenke’s theorem

Mathematica Montisnigri, Vol LVI, pp. 29‐34, (2023) DOI

In this paper, as an application of Zalcman’s lemma in $\mathbb{C}^{n}$, we give a sufficient condition for normality of holomorphic functions of several complex variables, which generalizes previous known one‐dimensional criterion of A.J. Lohwater and Ch. Pommerenke.

Applications of Zalcman’s Lemma in $\mathbb{C}^{n}$

In: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds) Current Trends in Analysis, its Applications and Computation. Trends in Mathematics (Proceedings of the 12th ISAAC Congress, Aveiro, Portugal, 2019). Birkhäuser, Cham. pp. 67‐74, (2022) DOI

The aim of this paper is to give some applications of Marty’s Criterion and Zalcman’s Rescalling Lemma.

On a normality criterion of P. Lappan

Revue Roumaine de Mathématiques Pures et Appliqué. LXVII(1‐2), 45‐49, (2022) PDF

In this paper, as an application of Zalcman’s lemma in $\mathbb{C}^{n}$ we give a sufficient condition for normality of a family of holomorphic functions of several complex variables, which generalizes previous known one‐dimensional results of H. L. Royden, W. Schwick, and P. Lappan.

Generalization of Lohwater‐Pommerenke’s
Theorem

arXiv.org (2023) DOI

In one‐dimensional case there are many criteria known for a meromorphic function to be normal, and the Lohwater and Pommerenke add a very valuable criterion to this list: a nonconstant function $f$ meromorphic in unit disc $U\subset \mathbb{C}$ is normal if and only if there do not exist sequences $\{z_{n}\}$ and $\{\rho_{n}\}$ with $z_{n}\in U$, $\rho_{n}>0$, $\rho_{n}\to 0$, such that $$\lim_{n\to \infty}f(z_{n}+\rho_{n}t)=g(t)$$ locally uniformly in $\mathbb{C}$, where $g(t)$ is a nonconstant meromorphic function in $\mathbb{C}$.

In this paper, as an application of Marty’s criterion in $\mathbb{C}^{n}$, we give a sufficient condition for normality of holomorphic functions of several complex variables, which generalizes previous known one‐dimensional criterion of A.J. Lohwater and Ch. Pommerenke.

Schottky’s theorem in $\mathbb{C}^{n}$

arXiv.org (2022) DOI

The aim of this note is to give a proof of the Schottky theorem in general domains in $\mathbb{C}^{n}$. The proof is short and works for the cases $n = 1$ and $n > 1$ at the same time.

On normal families in $\mathbb{C}^{n}$

Complex Var. Elliptic Equ. 67(1), 1‐8 (2022) DOI

We show that a family $\mathcal{F}:=\{f\}$ of functions holomorphic in a domain $\Omega\subset \mathbb{C}^{n}$ is normal if all eigenvalues of the complex Hessian matrix of $\ln(1+|f|^{2})$ are uniformly bounded away from zero on compact subsets of $\Omega \subset\mathbb{C}^{n}$.

Zalcman–Pang’s lemma in $\mathbb{C}^{n}$

Complex Var. Elliptic Equ. 66(12), 1991‐1997, (2021) DOI

The aim of this paper is to give a proof of Zalcman–Pang’s Rescalling Lemma in $\mathbb{C}^{n}$.

An improvement of Zalcman’s lemma in $\mathbb{C}^{n}$

Journal of Classical Analysis, 17(2), 109–118, (2021) DOI

The aim of this article is to improve the proof of Zalcman’s lemma in $\mathbb{C}^{n}$.

On a normality criterion of W. Schwick

J Geom Anal 31, 5355–5358 (2021). DOI

In this paper, as an application of Zalcman’s lemma in $\mathbb{C}^{n}$, we give a sufficient condition for normality of a family of holomorphic functions of several complex variables, which generalizes previous known one‐dimensional results of H.L. Royden and W. Schwick.

Zalcman’s lemma in $\mathbb{C}^{n}$

Complex Var. Elliptic Equ. 65(5), 796–800 (2020). DOI

The aim of this paper is to give a proof of Zalcman’s Rescalling Lemma in $\mathbb{C}^{n}$.

On normal families of holomorphic functions

Math. Montisnigri 36, 5‐13 (2016).

On admissible limits of holomorphic functions of several complex variables

Ann. Univ. Buchar., Math. Ser. 5(63), No. 1, 71‐82 (2014).

The aim of the present article is to establish the connection between the existence of the limit along the normal and the admissible limit at a fixed boundary point for holomorphic functions of several complex variables.

On a normality criterion of Mandelbrojt

Complex Var. Elliptic Equ. 59(10), 1388‐1394 (2014). DOI

Extension of the classical Mandelbrojt’s criterion for normality of a family of zero‐free holomorphic functions of several complex variables is given. We show that a family of holomorphic functions of several complex variables whose corresponding Levi form is uniformly bounded away from zero is normal.

Estimates for holomorphic functions with values in $\mathbb{C}\setminus\{0,1\}$

Advances in Pure Mathematics, 3(6), 586‐589, (2013) DOI

Extension of classical Mandelbrojt’s criterion for normality to several complex variables is given. Some inequalities for holomorphic functions which omit values $0$ and $1$ are obtained.

The Lindelöf principle in $\mathbb{C}^{n}$

Central European Journal of Mathematics, 11, 1763–1773, (2013) DOI

Review by Steven George Krantz (MR3080235):

The classical Lindelöf principle on the unit disc in the complex plane says that a bounded holomorphic function with radial boundary limit at a boundary point $p$ also has nontangential boundary limit at $p$.

In this paper, Dovbush explores generalizations of this result in both one and several complex variables. He obtains particularly sharp results on the polydisc.

The Lindelöf principle for holomorphic functions of infinitely many variables

Complex Var. Elliptic Equ. 56, 2011 ‐ Issue 1‐4: Dedicated to Professor Chung‐Chun Yang, 315‐323, (2011) DOI

Let $D$ be a convex bounded domain in a complex Banach space. A holomorphic function $f \colon D \rightarrow \mathbb{C}$ is called a normal function if the family $\mathcal{F}_{f}= \{f \circ \varphi : \varphi \in \mathcal{O}(\Delta, D)\}$ forms a normal family in the sense of Montel (here $\mathcal{O}(\Delta, D)$ denotes the set of all holomorphic maps from the complex unit disc into $D$). Let $\{x_{n}\}$ be a sequence of points in $D$ which tends to a boundary point $\xi \in \partial D$, such that $\lim_{n\to \infty}f(x_{n}) = l$ for some $l\in \overline{\mathcal{C}}$. The sufficient conditions on a sequence $\{x_{n}\}$ of points in $D$ and a normal holomorphic function $f$ are given for $f$ to have the admissible limit value $l$, thus extending the result obtained by Bagemihl and Seidel. This result is used to draw a new Lindelöf principle in the holomorphic function theory of infinite many variables. The results in this article are improvements of earlier results of Čirka, Dovbush, Cima and Krantz.

On the Lindelöf‐Gehring‐Lohwater theorem

Complex Var. Elliptic Equ. 56(5), 417‐421, (2011) DOI

In $\mathbb{C}^{n}$ we give an extension of the Lindelöf‐Gehring–Lohwater theorem involving two paths. A classical theorem of Lindelöf asserts that if $f$ is a function analytic and bounded in the unit disc $U$ which has the asymptotic value $L$ at a point $\xi\in \partial U$ then it has the angular limit $L$ at $\xi$. Later Lehto and Virtanen proved that a normal function $f$ has at most one asymptotic value at any given point $\xi\in \partial U$. Subsequently, the hypothesis of the existence of an asymptotic value has been weakend by Gehring and Lohwater. In this paper we extend their results to the higher dimensional case.

Boundary behaviour of Bloch functions and normal functions

Complex Var. Elliptic Equ. 55, Issue 1‐3: A tribute to Prof. Dr. C. Andreian Cazacu on the occasion of her 80th birthday, 157‐166 (2009) DOI

The purpose of the present article is to give the version of the Lindelöf principle which is valid in bounded domains in with $C^{2}$‐smooth boundary. We also prove that if a Bloch function is bounded on a $\mathcal{K}$‐special curve ending at a given boundary point, it is bounded on any admissible domain with vertex at the same point.

​ On normal and non‐normal holomorphic functions on complex Banach manifolds

Ann. Scuola Norm. Sup. Pisa Cl. Sci (5) Vol. VIII, 2009, pp.1‐15. DOI

Let $X$ be a complex Banach manifold. A holomorphic function $f \colon X \to \mathbb{C}$ is called a normal function if the family $\mathcal{F}_{f}:= \{ f \circ \varphi : \varphi \in \mathcal{O}(\Delta, X)\}$ forms a normal family in the sense of Montel (here $\mathcal{O}(\Delta, X)$ denotes the set of all holomorphic maps from the complex unit disc into $X$). Characterizations of normal functions are presented. A sufficient condition for the sum of a normal function and non‐normal function to be non‐normal is given. Criteria for a holomorphic function to be non‐normal are obtained. These results are used to draw one interesting conclusion on the boundary behavior of normal holomorphic functions in a convex bounded domain $D$ in a complex Banach space $V$. Let $\{x_{n}\}$ be a sequence of points in $D$ which tends to a boundary point $\xi \in \partial D$ such that $\lim_{n\to \infty}f (x_{n}) = L$ for some $L \in \mathbb{C}$. Sufficient conditions on a sequence $\{x_{n}\}$ of points in $D$ and a normal holomorphic function $f$ are given for $f$ to have the admissible limit value $L$, thus extending the result obtained by Bagemihl and Seidel.

Boundary behaviour of normal functions

Progress in Analysis and its Applications, Proceedings of the 7th International Isaac Congress Imperial College, London, UK, July 13‐18, 2009, pp. 39‐44. DOI

In multidimensional case we give an extension of the Lindelöf‐Lehto‐Virtanen theorem for normal functions and Lindelöf‐Gehring‐Lohwater theorem involving two paths for bounded functions.

Bloch functions on complex Banach manifolds

Mathematical Proceedings of the Royal Irish Academy, 108A(1), 27–32, 2008. DOI

Let $X$ be a complex Banach manifold. A holomorphic function $f \colon X \to \mathbb{C}$ is called a Bloch function if the family $\mathcal{F}_{f}:= \{f \circ \varphi - f(\varphi(0)) : \varphi \in \mathcal{O}(\Delta, X)\}$ forms a normal family in the sense of Montel, where $\mathcal{O}(\Delta, X)$ denotes the set of holomorphic maps from the complex unit disc to $X$. In this paper Bloch functions on complex Banach manifolds are studied. The main result shows that many of the equivalent definitions of Bloch functions on the unit disk are also equivalent in the general setting.

On Bloch and normal functions on complex Banach manifolds

Proceedings of the $6$th International ISAAC Congress, Ankara, Turkey, August 13–18, 2007, pp. 122–131. DOI

Let $X$ be a complex Banach manifold. A holomorphic function $f \colon X \to \mathbb{C}$ is called a Bloch function (resp., a normal function) if the family $\mathcal{F}_{f}:= \{f \circ \varphi - f(\varphi(0)) : \varphi \in \mathcal{O}(\Delta, X)\}$ (resp., $\mathcal{F}:= \{f \circ \varphi: \varphi \in \mathcal{O}(\Delta, X)\}$) forms a normal family in the sense of Montel, where $\mathcal{O}(\Delta, X)$ denotes the set of holomorphic maps from the complex unit disc to $X$. Characterizations of normal and Bloch functions are presented. A sufficient condition for the sum of a normal function and a non‐normal function to be non‐normal is given. Criteria for a holomorphic function to be non‐normal are obtained.

On normal and non‐normal holomorphic functions on complex Banach manifolds

Proceedings of the $6$th Congress of Romanian mathematicians, Bucharest, Romania, June 28‐July 4, 2007, 145‐152. DOI

Existence of $\mathcal{K}$‐limits of holomorphic maps

Math Notes 77, 471–475 (2005). DOI

Let $D$ be a complete hyperbolic domain in $\mathbb{C}^{n}$, $n >1$, and $N$ a compact Hermitian manifold. We prove a criterion for the existence of the $\mathcal{K}$‐limit of an arbitrary holomorphic map $f \colon D \to N$ at an arbitrary boundary point $D$ under the condition that $f$ has the corresponding radial limit at this point.

On the existence of $\mathcal{K}$‐admissible limits of holomorphic maps

Opuscula Mathematica, 23, 15‐20, 2003.

Given a complete hyperbolic domain $D \subset \mathbb{C}^{n}$ and a holomorphic map $f \colon D \to \mathbb{C}$, we give a necessary and sufficient condition concerning the existence of $\mathcal{K}$‐admissible limit of $f$, if the corresponding radial limit exist.

​ $X$‐normal mappings

Bul. Acad. Ṣtiinṭe Repub. Mold. Mat., 2003, no. 3, 71–82 PDF

This is a survey of achievements in the theory of normal holomorphic mappings. We systematize and present all the results on the subject that are obtained by the author from the beginning of the theory until the date of writing the paper.

About negligible sets for normal mappings of several complex variables

Revue Roumaine de Mathématiques Pures et Appliqué. 46, 29‐45, 2001.

Suppose that $D$ is a domain in $\mathbb{C}^{n}$, $n>1$, $E \subset D$ is closed in $D$ and has zero $(2n-1)$‐dimensional Hausdorff measure. If $f\colon D\setminus E \to \overline{\mathbb{C}}$ is $C$‐normal, then $f$ extends to a holomorphic mapping $F\colon D\setminus E \to \overline{\mathbb{C}}$ which is $C$‐normal on $D$. We also point out that classes of $C$‐normal mappings and $\mathcal{K}$‐normal mappings (holomorphic mappings in the sense of J. A. Cima and S. G. Krantz [Duke Math. J. 50, 303–328 (1983; Zbl 0522.32003)]) are different.

(with V. I. Gavrilov) Normal mappings (a survey)

Math. Montisnigri, XIV, 5‐61, 2001.

The paper is a survey of achievements in the theory of normal holomorphic mappings. Some open problems are mentioned.

Polynomiality criterion for entire functions of several complex variables

Math Notes 66, 409–410 (1999). DOI

The “radial” polynomiality criterion for entire functions of several complex variables is proved.

On the existence of admissible and weakly admissible limits of functions of several complex variables

Sib. Math. J. 33, No. 4, 737‐739 (1992); translation from Sib. Mat. Zh. 33, No. 4, 212‐214 (1992). Siberian Math. J., 33:4 (1992), 737‐739 DOI

Admissible limits of normal holomorphic functions of several complex variables

Mathematical Notes of the Academy of Sciences of the USSR 47, 449–453 (1990). DOI

Lindelöf’s theorem in $\mathbb{C}^{n}$

Ukr. Math. J. 40, No. 6, 673‐676 (1988); translation from Ukr. Mat. Zh. 40, No. 6, 796‐799 (1988). DOI

On the existence of admissible limits of functions of several complex variables

Siberian Math. J., 28:3 (1987), 411–414 DOI

Boundary behavior of holomorphic functions of several complex variables

Math. Notes, 39:3 (1986), 196–199 DOI

We deal with a proven theorem in which one gets sufficient conditions for two limiting values of a function which is bounded and holomorphic in the unit disc $U$, corresponding to two different sequences of points of $U$, to be equal (see W. Seidel On the cluster values of analytic functions, Trans. Amer. Math. Soc. 34 (1932), no. 1, 1–21 DOI) In this paper we prove a generalization of this theorem of Seidel to the case of bounded holomorphic functions in strictly pseudoconvex domains of the space $\mathbb{C}^{n}$.

(with V. I. Gavrilov) Boundary singularities generated by cluster sets of functions of several complex variables

Sov. Math., Dokl. 26, 186‐189 (1982); translation from Dokl. Akad. Nauk SSSR 265, 1047‐1050 (1982). PDF

Boundary behavior of normal holomorphic functions of several complex variables

Sov. Math., Dokl. 25, 267‐270 (1982); translation from Dokl. Akad. Nauk SSSR 263, 14‐17 (1982). PDF

Review by T. J. Barth (Zbl 0531.32003):

Let $D$ be a strictly pseudoconvex bounded domain in $\mathbb{C}^{n}$ $(n>1)$, and let $f$ be a holomorphic function on $D$. By a $P$‐sequence for $f$ the author means a sequence in $D$ converging to a boundary point in such a way that $f$ takes on every complex value (with at most one exception) infinitely often on every Bergman metric $\varepsilon$‐neighborhood of every subsequence. The function $f$ is called normal on $D$ if it does not possess a $P$‐sequence. The author proves the following criterion: $f$ is normal if and only if $|\nabla f(z)|=O(1+|f(z)|^{2})$ as $z$ approaches the boundary of D; here $|\nabla f|$ denotes the modulus of the gradient of $f$ measured in the Bergman metric of $D$. This leads to an analogue of a result about Fatou points first proved for the disk by F. Bagemihl and W. Seidel [Comment. Math. Helv. 36, 9‐18 (1961; Zbl 0125.317)]: if f is normal, then f has admissible limits (within certain Stoltz‐type regions) on an everywhere dense subset of the boundary of D.

Lindelöf’s theorem in $\mathbb{C}^{n}$

Mosc. Univ. Math. Bull. 36, No. 6, 41‐44 (1981); translation from Vestn. Mosk. Univ., Ser. I 1981, No. 6, 33‐36 (1981). PDF (in Russian)

Review by Harold P. Boas (MR648587):

The theorem of Lindelöf in question says that if a bounded holomorphic function $f$ in the unit disk has a limit along some path (for instance a radius) terminating at a boundary point $p$, then $f$ has a nontangential limit at $p$, that is a limit within any cone lying in the disk and with vertex at $p$. For smooth domains in $\mathbb{C}^{n}$ the analogous theorem holds, and one can even supplant cones with the wider class of so‐called admissible approach regions, allowing tangential approach to the boundary in certain directions [see E. M. Chirka, Mat. Sb. 92(134) (1973), 622‐644; MR0338415; E. M. Stein, Boundary behavior of holomorphic functions of several complex variables, Princeton Univ. Press, Princeton, N.J., 1972; MR0473215]. The author restricts to the class of strongly pseudoconvex domains in $\mathbb{C}^{n}$ but extends the class of functions from bounded holomorphic functions to so‐called normal holomorphic functions; in this setting the author proves that if $f$ has a limit along one nontangential path terminating at a boundary point $p$ then $f$ has a limit at $p$ within any admissible region.

Normal functions of many complex variables

Mosc. Univ. Math. Bull. 36, No. 1, 44‐48 (1981); translation from Vestnik Moskov. Univ., Ser. I Mat. Mekh. 1981, no. 1, 38–32. PDF (in Russian)

English translation (MR613124):

The normal meromorphic functions of one complex variable introduced by O. Lehto and K. I. Virtanen [Acta Math. 97 (1957), 47–65; MR0087746] have been studied by many authors. A fairly complete survey of results can be found in A. J. Lohwater’s paper [Mathematical analysis, Vol. 10 (Russian), pp. 99–259, Akad. Nauk SSSR VINITI, Moscow, 1973; MR0399467]. In the present paper we introduce the notion of a normal holomorphic function of several complex variables and prove two criteria for normality of the function in domains of a special type. The proof is based on the criterion of normality of families of holomorphic functions of several complex variables. For the one‐dimensional case this was established by F. Marty [Ann. Fac. Sci. Univ. Toulouse (3) 23 (1931), 183–261; Zbl 4, 118].

Conferences and talks

A nulset for normal functions in several variables

International conference on Complex Analysis and related topics. Iassy. The VIII Romanian‐Finnish seminar. Romania. August 22–27, 1999, pp. 25‐26.

Normal function in $\mathbb{C}^{n}$: a survey

International conference on Complex Analysis and related topics. The IX Romanian‐Finnish seminar. Brashov, Romania, August 27‐31, 2001, p. 24.

Extension properties of $X$‐normal mappings

The 5th congress of Romanian mathematicians. Piteşti. Romania. June 22 – 28, 2003, p. 44‐45.

On $X$‐normal holomorphic functions on a complex Banach manifold

International conference on Complex Analysis and related topics. The X‐Romanian‐Finnish seminar. Cluj‐Napoca, Romania, August 14‐19, 2005, p. 25.

Bloch functions on complex Banach manifolds

International conference on Complex Function Theory and Application ‐ Brasov, Romania,1‐5 September 2006.

On normal and non‐normal holomorphic functions on complex Banach manifolds

The $6$th Congress of Romanian mathematicians. June 28 ‐ July 4, 2007, Bucharest, Romania.

On Bloch functions and normal functions on complex Banach manifolds

The $6$th International ISAAC Congress. Ankara, Turkey, 13‐18 August 2007.

Normality and $P$‐point sequences

Exploratory Workshop on Recent trends in complex analysis and related topic Alba Iulia, Romania, August 14‐16, 2008, pp. 8‐9.

Normality and $P$‐point sequences

International Conference on Complex Analysis and Related Topics, The 11‐th Romanian‐Finnish Seminar, Alba Iulia, Romania, August 14‐19, 2008, pp. 22‐23.

On the Lindelöf theorem

International Conference on Complex Analysis and Related Topics, The 12th Romanian‐Finnish Seminar, 2009, pp. 19‐20. DOI

Admissible and weakly admissible limit theorems

X‐th Conference On Analytic Functions And Related Topics, June 26 – 29, 2011 Chełm (Poland)

Zalcman’s lemma in $\mathbb{C}^{n}$

The Ninth Congress of Romanian Mathematicians June 28 ‐ July 3, 2019, Galați, Romania

The aim of this talk is to give a proof of Zalcman’s Rescaling Lemma in $\mathbb{C}^{n}$, $n \geq 1$. We also give some applications of Zalcman’s Rescaling Lemma in $\mathbb{C}^{n}$.

Zalcman’s lemma in $\mathbb{C}^{n}$

12th International ISAAC Congress July 29‐2, 2019, Aveiro, Portugal