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 $\mathbf{C}^{n}$ 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

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

(joint work with S.G.Krantz)
Com­plex Var. El­lip­tic Equ. 1‐11. 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)
Book will be pub­lished on Feb­ru­ary 16, 2025 (Pre‐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 $\mathbf{C}$ 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 $\mathbf{C}^{n}$

(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),

• •

  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 $\mathbf{C}^{n}$, 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 $\mathbf{C}^{n}$

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 $\mathbf{C}^{n}$ 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 $U\subset \mathbf{C}$ 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 $\mathbf{C}$, where $g(t)$ is a non­con­stant mero­mor­phic func­tion in $\mathbf{C}$.

In this pa­per, as an ap­pli­ca­tion of Mar­ty’s cri­te­rion in $\mathbf{C}^{n}$, 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 $\mathbf{C}^{n}$

arXiv.org (2022) DOI

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

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

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

We show that a fam­ily $\mathcal{F}:=\{f\}$ of func­tions holo­mor­phic in a do­main $\Omega\subset \mathbf{C}^{n}$ 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 $\Omega \subset\mathbf{C}^{n}$.

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

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 $\mathbf{C}^{n}$.

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

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 $\mathbf{C}^{n}$.

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 $\mathbf{C}^{n}$, 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 $\mathbf{C}^{n}$

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 $\mathbf{C}^{n}$.

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 $\mathbf{C}\setminus\{0,1\}$

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 $\mathbf{C}^{n}$

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 $f \colon D \rightarrow \mathbf{C}$ 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 $\mathbf{C}^{n}$ 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 $f \colon X \to \mathbf{C}$ 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 $L \in \mathbf{C}$. 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 $f \colon X \to \mathbf{C}$ 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 $f \colon X \to \mathbf{C}$ 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 $\mathbf{C}^{n}$, $n >1$, 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 $D \subset \mathbf{C}^{n}$ and a holo­mor­phic map $f \colon D \to \mathbf{C}$, 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 $\mathbf{C}^{n}$, $n>1$, $E \subset D$ is closed in $D$ and has zero $(2n-1)$‐di­men­sional Haus­dorff mea­sure. If $f\colon D\setminus E \to \overline{\mathbf{C}}$ is $C$‐nor­mal, then $f$ ex­tends to a holo­mor­phic map­ping $F\colon D\setminus E \to \overline{\mathbf{C}}$ 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 $\mathbf{C}^{n}$

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 $\mathbf{C}^{n}$.

(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 $\mathbf{C}^{n}$ $(n>1)$, 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 $\mathbf{C}^{n}$

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 $\mathbf{C}^{n}$ 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 $\mathbf{C}^{n}$ 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 $\mathbf{C}^{n}$: 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 $\mathbf{C}^{n}$

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 $\mathbf{C}^{n}$, $n \geq 1$. We also give some ap­pli­ca­tions of Zal­cman’s Rescal­ing Lemma in $\mathbf{C}^{n}$.

Zal­cman’s lemma in $\mathbf{C}^{n}$

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