# Anatolii Asirovich Goldberg

### Born: 2 April 1930 in Kiev, Ukraine

Died: 11 October 2008 in Netanya, Israel

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**Anatolii Asirovich Goldberg**'s father, Asher Goldberg, was a medical doctor while his mother was a secondary school teacher. As the name Goldberg suggests, the family was Jewish. Anatolii Asirovich spent the first three years of his life in Kiev before the family moved to Zaporizhzhy, a city in south eastern region of Ukraine on the Dnieper River. The town had grown up in the years just before the Goldberg family moved there due to the construction of the Dnieper hydroelectric station in 1927-32. Anatolii Asirovich was nine years old when World War II began but initially this did not affect the young boy.

The Molotov-Ribbentrop non-aggression pact between Germany and the Soviet Union meant that the initial years of the war had little effect on life in Zaporizhzhy and Anatolii Asirovich was able to continue his schooling. However, things changed dramatically on 22 June 1941 when Germany broke the non-aggression pact and invaded the Soviet Union. The German armies advanced across the Ukraine and, although Zaporizhzhy was in eastern Ukraine, the German forces were approaching the city towards the end of 1941. Goldberg's mother fled with her son to the east of the Soviet Union where they remained safe for the duration of the war. When the war ended, the Goldberg family settled in Lviv. This city had been part of Poland before World War II began (known as Lvov) but, after being occupied by German troops during the war, was annexed by the Soviets in 1945. Goldberg completed his secondary school studies in Lviv. There he showed remarkable mathematical talents, winning first place in the Mathematical Olympiad competition of 1947. Aleksandr Sergeevich Kovanko (1893-1975) was chairman of the local Olympiad committee and he encouraged Goldberg to aim for a career as a university mathematics teacher.

The University of Lviv reopened as a Ukrainian university in 1945 and several leading mathematicians were appointed there. Aleksandr Sergeevich Kovanko, who we just mentioned, had been a student of Dimitri Fedorovich Egorov and had participated in Nikolai Nikolaevich Luzin's famous school at the University of Moscow in the 1920s. His main interests were almost periodic functions, the problem of surface area, and sequences of functions. Ivan Georgievich Sokolov (1903-1993) held the Chair of the Theory of Functions and Probability Theory at Lviv. Several other mathematicians at the university were equally important for Goldberg's development, including Boris Vladimirovich Gnedenko, Yaroslav Borisovich Lopatynsky, who held the chair of differential equations, and Lev Israelevich Volkovyskii who had been a student of Mikhail Alekseevich Lavrent'ev and was working on complex analysis, particularly quasiconformal mappings and Riemann surfaces.

In both [2] and [3], Goldberg's own description of his time as a student at Lviv is recorded. We give a version below:-

Goldberg wrote his Master's thesis advised by Volkovyskii. In it he studied the inverse problem of the theory of the distribution of values which had been proposed by Nevanlinna in the 1920s. He was awarded the degree in 1952 for the thesisAt university, it was Professors Lev Israelevich Volkovyskii and Ivan Georgievich Sokolov who had the greatest influence on me. This applies to both their mathematical influence and their civic position. It must be said that in those years a serious price could be paid for such a bold civic stand. I was a troublesome student and I was twice expelled from the Young Communist League and from the university but later reinstated. The offences were of a political nature: "bourgeois liberalism", "loss of Young Communist vigilance", and so on. During my first and second years in the University, I took part in a seminar on the book by Polya and Szego, conducted by Lev Israelevich Volkovyskii and Ivan Georgievich Sokolov. During my second year the seminar divided. I went to Sokolov and began to interest myself in the constructive theory of functions. I greatly enjoyed this activity; at that time I P Natanson's book had come out and it was clearly and simply written. From my fellow students who went to study under Volkovyskii, I learnt that their fundamental text was Rolf Nevanlinna's monograph "Single-valued analytic functions", of which the only copy in Lviv was Volkovyskii's. To study this book, one had to go to his apartment at prescribed times. Nevanlinna's book was difficult even for senior students and even worse for second year students, who had still not studied the theory of functions of a complex variable. Hearing their complaints, I thanked my lucky stars that I had distanced myself from that accursed book. But here chance intervened. Looking through the catalogue of the Lviv Regional Library I saw a card for Nevanlinna's book in the section "Analytic geometry". I could not put the treasure down, ordered the book on my ticket, deserted and defected to the Volkovyskii camp.

*On a problem in the theory of distribution of values of meromorphic functions*. He published the results of his investigations in five papers in 1954, one in Ukrainian and one in Russian with the same title as his thesis, and the other papers in Russian: (with P P Belinskii)

*Application of a theorem on conformal mappings to questions of invariance of defects of meromorphic functions*;

*On the influence of clustering of algebraic branch points of a Riemann surface on the order of growth of a meromorphic mapping function*; and

*On defects of meromorphic functions*. We note that the results that Goldberg obtained for the inverse problem of the theory of the distribution of values which he published in 1954 were not improved on during the next twenty years.

Anti-Semitism in the Ukraine at this time meant that, although Goldberg would have wished to work towards his Candidate's degree (equivalent to a Ph.D.) by studying at a university, this was almost impossible. He therefore took a position as a secondary school teacher of mathematics in the village of Zabolotsy. When Stalin died in 1953 Goldberg thought that the position of Jews might improve so he applied to study for his candidate's Degree at Lviv University. However, his application was rejected. There was, however, another possible, but extremely difficult, route to a Candidate's Degree left open to Goldberg. Even though he could not register as a student, nevertheless he was allowed to sit the qualifying examinations and submit a thesis. Goldberg undertook the necessary work for his Candidate's Degree while working as a school teacher, teaching for 30 hours a week and undertaking further administrative tasks. He submitted his thesis to Lviv University in 1955 and this not only led to the award of the degree but, as we shall explain, it led to Goldberg collaborating with the mathematicians at Kharkov University.

Boris Yakovlevich Levin examined Goldberg's Candidate's Thesis

*Some Problems of Distribution of Values of Meromorphic Functions*in 1955 at Lviv University. This led to Goldberg collaborating with Levin's colleagues at Kharkov University, in particular with Iossif Vladimirovich Ostrovskii. The collaboration between Goldberg and Ostrovskii led to important results. Their first joint papers were

*Some theorems on the growth of meromorphic functions*(Russian) (1961) and

*New investigations on the growth and distribution of values of entire and meromorphic functions of genus zero*(Russian) (1961). Another joint paper

*Application of a theorem of W Hayman to a question in the theory of expansions of probabilistic laws*(Russian) (1967) was followed by the classic joint monograph

*The distribution of values of meromorphic functions*(Russian) (1970). Here is a short extract of a review of this book by Walter Hayman:-

Goldberg and Ostrovskii explain their aims in writing the monograph in a Preface:-Nevanlinna theory has made considerable progress in the50s and60s, progress to which the authors of the present book have made distinguished contributions. Their book is detailed and authoritative, and there is little of importance in the subject of value distribution of functions meromorphic in the plane that is missing from this book and that cannot be found in the earlier books on the subject. The subject matter is illustrated at every stage by subtle and detailed examples -some of which are here published for the first time and some of which are not easily accessible elsewhere. ... All function theorists are indebted to the authors for this comprehensive and scholarly work.

We should return to 1955, the year in which Goldberg was awarded his Candidate's Degree from Lviv University. Following the award of this degree, Goldberg was appointed to Uzhgorod University as a docent where he worked for eight years before returning to Lviv University in 1963. Two years later, in 1965, he submitted his doctoral thesis (equivalent to the habilitation or to a D.Sc.) entitledIn this book ... the main attention is concentrated on the problems internal to the value distribution theory, which include the following problems:(i)To what extent the main inferences of Nevanlinna's theory have final character and cannot be improved further;(ii)What properties of Picard's exceptional values are preserved for a wider class of exceptional values considered in the value distribution theory;(iii)Which are the connections between Nevanlinna's characteristics and other quantities characterizing asymptotic properties of entire and meromorphic functions;(iv)Study of asymptotic properties and value distribution of meromorphic functions belonging to some special classes which are on one hand sufficiently narrow to give new information not implied by general theorems, and on the other hand sufficiently wide for being of interest for the general theory;(v)Study of the value distribution with respect to arguments(not only with respect to moduli as in classical Nevanlinna theory). We pay great attention to examples of functions with "exotic" properties. Without them the reader will get a restricted image of the theory under consideration. Examples of functions having unusual properties play in the theory a role as important as counterexamples do in Real Analysis.

*The distribution of values and asymptotic properties of entire and meromorphic functions*(Russian). The authors of [2] write:-

After submitting this thesis, he was promoted to full professor at Lviv and he held this position until 1997. The year he was made a full professor, he [3]:-Let us give a catalogue of the investigative strands to be found in Goldberg's thesis:(1)

to what extent are the fundamental conclusions of Nevanlinna's theory best possible and not capable of refinement;(2)

which properties of Picard's exceptional values are retained by defect values;(3)

what sort of link is there between the characteristics of Nevanlinna's theory and other magnitudes that describe asymptotic properties of the function;(4)

what sort of link is there between defect values and the distribution of a-points with respect to arguments;(5)

asymptotic properties and distributions of the values of various specific classes of entire and meromorphic functions.

Goldberg kept up a remarkable publication record with over 150 items listed in [3] with around twenty further items published after the article [3] appeared. As examples of these papers, we mention... began his seminar at the Mathematics Department of Lviv University, on Tuesdays for two hours. Almost all results of Lviv mathematicians related to the theory of entire and meromorphic functions were thoroughly considered and discussed at the seminar.

*Estimates of conformal maps of curvilinear strips*(2002). Dov Aharonov writes in a review:-

Goldberg was asked to give the opening memorial plenary lecture on memorial meeting in honour of Shlomo Strelitz at the 'Conference of Differential Equations and Complex Analysis' at the University of Haifa in December 2000. He gave the lectureThe author is interested in the question of conformality of a map at a boundary point. Following previous work of himself and others, the author uses a strip-like canonical domain. ... In this paper the author does not prove conditions for the existence of the angular derivative but rather obtains rough estimates for a wider class of strips.

*On the growth of entire solutions of algebraic differential equations*which was published in 2005. Also in 2005 his paper

*On a connection between the number of poles of a meromorphic function and the number of zeros of its derivatives*was published.

Goldberg married twice. The authors of [3] state:-

In 1997 Goldberg retired from his professorship at Lviv and took up a position at Bar Ilan University in Israel. This university is situated in Ramat Gan, east of Tel Aviv-Yafo. He lived in Netanya on the Mediterranean coast, 30 km north of Tel Aviv-Yafo.In1964Goldberg married Basya B Lekhtman. They have three sons, Alexander, Mikhael and Victor. Goldberg brings up a son, Mark, from a previous marriage.

He received several honours for his outstanding contributions. In 1992 Goldberg, Ostrovskii and Levin, were joint recipients of the State Prize of Ukraine. He was also honoured by having mathematical terms named after him. However, there appears to be some confusion in the literature regarding which things named for Goldberg are actually named for Anatolii Asirovich Goldberg. The problem arises since there are a number of mathematicians named Goldberg. However, the 'Goldberg constant' is name after the subject of this biography and appears in his paper

*A certain theorem of Landau type*(1973). He was also honoured following his death with a 'Conference on complex analysis dedicated to the memory of Anatolii Asirovich Goldberg' being held in Lviv, Ukraine, from 31 May to 5 June 2010. It was organised by the Ivan Franko National University of Lviv and the Lviv Mathematical Society.

In [3] the authors describe Goldberg's love of writing:-

With many of his colleagues Goldberg keeps intensive correspondence. Besides mathematics, his letters contain concise reports on current economical and political developments, local news, short reminiscences, interesting historical remarks, everything flavoured with a wonderful humour. Goldberg's talent as a remarkable storyteller goes hand in hand with the literary quality of his letters.

**Article by:** *J J O'Connor* and *E F Robertson*

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