Pe­ter Dovbush

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 Cn\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

dr hab. P. V. Dovbush

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 Cn\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

(sub­mit­ted)

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 ff mero­mor­phic in unit disc UCU\subset \mathbf{C} is nor­mal if and only if there do not ex­ist se­quences {zn}\{z_n\} and {ρn}\{\rho_n\} with znU,z_n\in U, ρn>0,\rho_n>0, ρn0,\rho_n\to 0, such that limnf(zn+ρnt)=g(t)\lim_{n\to \infty}f(z_n+\rho_nt)=g(t) lo­cally uni­formly in C,\mathbf{C}, where g(t)g(t) is a non­con­stant mero­mor­phic func­tion in C.\mathbf{C}.

In this pa­per, as an ap­pli­ca­tion of Mar­ty’s cri­te­rion in Cn\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.

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

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

We show that a fam­ily F:={f}\mathcal F:=\{f\} of func­tions holo­mor­phic in a do­main ΩCn\Omega\subset \mathbf C^n is nor­mal if all eigen­val­ues of the com­plex Hes­s­ian ma­trix of ln(1+f2)\ln(1+|f|^2) are uni­formly bounded away from zero on com­pact sub­sets of ΩCn.\Omega \subset\mathbf C^n.

Zal­cman–Pang’s lemma in Cn\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 Cn.\mathbf C^n.

An im­prove­ment of Zal­cman’s lemma in Cn\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 Cn.\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 Cn,\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 Cn\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 Cn.\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 C{0,1}\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 00 and 11 are ob­tained.

The Lin­delöf prin­ci­ple in Cn\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 pp also has non­tan­gen­tial bound­ary limit at p.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 DD be a con­vex bounded do­main in a com­plex Ba­nach space. A holo­mor­phic func­tion f​:DCf \colon D \rightarrow \mathbf C is called a nor­mal func­tion if the fam­ily Ff={fφ:φO(Δ,D)}\mathcal{F}_f = \{f \circ \varphi : \varphi \in \mathcal{O}(\Delta, D)\} forms a nor­mal fam­ily in the sense of Mon­tel (here O(Δ,D)\mathcal O(\Delta, D) de­notes the set of all holo­mor­phic maps from the com­plex unit disc into DD). Let {xn}\{x_n \} be a se­quence of points in DD which tends to a bound­ary point ξD,\xi \in \partial D, such that limnf(xn)=l\lim_{n\to \infty} f(x_n ) = l for some lC.l\in \overline{\mathcal{C}}. The sufficient con­di­tions on a se­quence {xn}\{x_n \} of points in DD and a nor­mal holo­mor­phic func­tion ff are given for ff to have the ad­mis­si­ble limit value l,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 Cn\mathbf{C}^n we give an ex­ten­sion of the Lin­delöf‐Gehring–Lo­hwa­ter the­o­rem in­volv­ing two paths. A clas­si­cal the­o­rem of Lin­delöf as­serts that if ff is a func­tion an­a­lytic and bounded in the unit disc UU which has the as­ymp­totic value LL at a point ξU\xi\in \partial U then it has the an­gu­lar limit LL at ξ.\xi. Later Lehto and Vir­ta­nen proved that a nor­mal func­tion ff has at most one as­ymp­totic value at any given point ξU.\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 C2C^2‐smooth bound­ary. We also prove that if a Bloch func­tion is bounded on a K\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 X be a com­plex Ba­nach man­i­fold. A holo­mor­phic func­tion f​:XCf \colon X \to \mathbf{C} is called a nor­mal func­tion if the fam­ily Ff:={fφ:φO(Δ,X)}\mathcal F_f := \{ f \circ \varphi : \varphi \in \mathcal{O}(\Delta, X)\} forms a nor­mal fam­ily in the sense of Mon­tel (here O(Δ,X)\mathcal{O}(\Delta, X) de­notes the set of all holo­mor­phic maps from the com­plex unit disc into XX). 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 DD in a com­plex Ba­nach space V.V. Let {xn}\{x_n\} be a se­quence of points in DD which tends to a bound­ary point ξD\xi \in \partial D such that limnf(xn)=L\lim_{n\to \infty} f (x_n) = L for some LC.L \in \mathbf{C}. Sufficient con­di­tions on a se­quence {xn}\{x_n\} of points in DD and a nor­mal holo­mor­phic func­tion ff are given for ff to have the ad­mis­si­ble limit value L,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 XX be a com­plex Ba­nach man­i­fold. A holo­mor­phic func­tion f​:XCf \colon X \to \mathbf{C} is called a Bloch func­tion if the fam­ily Ff:={fφf(φ(0)):φO(Δ,X)}\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 O(Δ,X)\mathcal O (\Delta, X) de­notes the set of holo­mor­phic maps from the com­plex unit disc to X.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 66th In­ter­na­tional ISAAC Con­gress, Ankara, Turkey, Au­gust 13–18, 2007, pp. 122–131.

Let XX be a com­plex Ba­nach man­i­fold. A holo­mor­phic func­tion f​:XCf \colon X \to \mathbf{C} is called a Bloch func­tion (resp., a nor­mal func­tion) if the fam­ily Ff:={fφf(φ(0)):φO(Δ,X)}\mathcal F_f := \{f \circ \varphi - f(\varphi(0)) : \varphi \in \mathcal{O}(\Delta, X)\} (resp., F:={fφ:φO(Δ,X)} \mathcal F := \{f \circ \varphi: \varphi \in \mathcal{O}(\Delta, X)\}) forms a nor­mal fam­ily in the sense of Mon­tel, where O(Δ,X)\mathcal{O}(\Delta, X) de­notes the set of holo­mor­phic maps from the com­plex unit disc to X.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 66th Con­gress of Ro­man­ian math­e­mati­cians, Bucharest, Ro­ma­nia, June 28‐July 4, 2007, 145‐152. DOI

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

Math Notes 77, 471–475 (2005). DOI

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

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

Opus­cula Math­e­mat­ica, 23, 15‐20, 2003.

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

X 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 DD is a do­main in Cn,\mathbf{C}^n , n>1,n>1, EDE \subset D is closed in DD and has zero (2n1)(2n-1)‐di­men­sional Haus­dorff mea­sure. If f​:DECf\colon D\setminus E \to \overline{\mathbf{C}} is CC‐nor­mal, then ff ex­tends to a holo­mor­phic map­ping F​:DECF\colon D\setminus E \to \overline{\mathbf{C}} which is CC‐nor­mal on D.D. We also point out that classes of CC‐nor­mal map­pings and K\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 Cn\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,U, cor­re­spond­ing to two differ­ent se­quences of points of U,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 Cn.\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 DD be a strictly pseudo­con­vex bounded do­main in Cn\mathbf{C}^n (n>1),(n>1), and let ff be a holo­mor­phic func­tion on D.D. By a PP‐se­quence for ff the au­thor means a se­quence in DD con­verg­ing to a bound­ary point in such a way that f f takes on every com­plex value (with at most one ex­cep­tion) infinitely of­ten on every Bergman met­ric ε\varepsilon‐neigh­bor­hood of every sub­se­quence. The func­tion ff is called nor­mal on DD if it does not pos­sess a PP‐se­quence. The au­thor proves the fol­low­ing cri­te­rion: ff is nor­mal if and only if f(z)=O(1+f(z)2)|\nabla f(z)|=O(1+|f(z)|^2) as zz ap­proaches the bound­ary of D; here f|\nabla f| de­notes the mod­u­lus of the gra­di­ent of ff mea­sured in the Bergman met­ric of D.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 Cn\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 ff in the unit disk has a limit along some path (for in­stance a ra­dius) ter­mi­nat­ing at a bound­ary point p,p, then ff has a non­tan­gen­tial limit at p,p, that is a limit within any cone ly­ing in the disk and with ver­tex at p.p. For smooth do­mains in Cn\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 Cn\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 ff has a limit along one non­tan­gen­tial path ter­mi­nat­ing at a bound­ary point pp then ff has a limit at pp 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 Cn\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 XX‐nor­mal map­pings

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

On XX‐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 66th Con­gress of Ro­man­ian math­e­mati­cians. June 28 ‐ July 4, 2007, Bucharest, Ro­ma­nia.

On Bloch func­tions and nor­mal func­tions on com­plex Ba­nach man­i­folds

The 66th In­ter­na­tional ISAAC Con­gress. Ankara, Turkey, 13‐18 Au­gust 2007.

Nor­mal­ity and PP‐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 PP‐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 Cn\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 Cn,\mathbf{C}^n, n1.n \geq 1. We also give some ap­pli­ca­tions of Zal­cman’s Rescal­ing Lemma in Cn.\mathbf{C}^n.

Zal­cman’s lemma in Cn\mathbf{C}^n

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