# Jovan Karamata

### Quick Info

Zagreb, Austria-Hungary (now Serbia)

Geneva, Switzerland

**Jovan Karamata**was an outstanding mathematician who lived through the wars of the first half of the 20th century in the Balkans. He gained international fame with a 2-page proof of the Hardy-Littlewood theorem, and also did excellent work on slowly varying functions and on Tauberian theory.

### Biography

**Jovan Karamata**was the son of Stevan Karamata (1866-1940) and his wife Desanka Vukomanovic (1870-1910). Stevan Karamata, who came from a famous family, was the fourth child of Atanasije Karamata and his wife Marija Jovica. He became a founder member of the Serbian Bank in Zagreb. This bank was very successful and he became director of a branch of the bank which was established in Budapest. We should mention that Stevan had a brother Kosta Karamata (1862-1920) who was a gymnasium mathematics teacher in Zemun. Kosta was a fine teacher, but also a research mathematician publishing three papers in the journal of the Yugoslav Academy of Sciences and Arts in Zagreb. Another of Stevan's brothers, also named Jovan Karamata, was the owner of a famous printing house, the director of the savings bank in Zemun, and a member of the Hungarian Parliament. Stevan Karamata married Desanka Vukomanovic and they had six children: Atanasije, Ozren, Srdan, Smiljka, Kosta, and Jovan, the subject of this biography. Atanasije died as a child so Jovan had four older siblings.

In 1909 Jovan Karamata began his primary schooling in Budapest, but soon his family moved to Zemun where there had been a Karamata family home for several generations. There he continued his primary schooling, completing it in 1912. By this time, however, Jovan's mother had died and Marija Karamata, his fraternal grandmother (by this time a widow), took on the role of mother to all of Stevan's children. "Nanna" Marija [5]:-

... the devoted and deeply pious woman, about whom Jovan Karamata spoke with great respect, had, according to him, a lasting influence on his spiritual development.Jovan began his secondary education at the Gymnasium in Zemun in 1912. Had the world situation been very different, his school education would almost certainly have been completed in Zemun, a town he always considered his home town which he dearly loved. The First Balkan War broke out in 1912 and involved the four Balkan states successfully defeating the Ottoman Empire. This had a relatively minor effect on Jovan's schooling in Zemun but the Second Balkan War, which broke out in 1913 worried Jovan's family much more. This war, a direct consequence of the various countries being dissatisfied with the treaty following the First Balkan War, led to Bulgaria attacking Serbia and Greece, while Romania attacked Bulgaria. Zemun, in Austro-Hungary at this stage, was considered in danger of Serbian attack, so Jovan's family sent the young boy to Osijek before the outbreak of World War I. Indeed Zemun did fall to the Serbian forces in August 1914, only a couple of weeks after World War I broke out. In 1915 Jovan's family moved him again, still trying to avoid the areas of conflict, and he went to Susak near Rijeka.

Not surprisingly, Jovan found these years extremely difficult and his education was suffering; he came close to having to repeat his third year at secondary school. His father, Stevan Karamata, worried about his children's education and also about the uncertainty caused by the fighting of World War I, decided later in 1915 to take his children to Switzerland to give them a better chance. Srdan, Smiljka, Kosta and Jovan went to Switzerland at this time but Jovan's brother Ozren only joined them at the beginning of 1917. Jovan spent one year studying at a boarding school in Cressier in the Canton of Bern where he became fluent in French. In 1916 he enrolled in a Gymnasium in Lausanne. Only 14 years old, he was alone at this time, living in a boarding house in the Vidy area of Lausanne which today is the headquarters of the International Olympic Committee. Karamata graduated from the Gymnasium in 1918 and, later that year, began to study at the Gymnase Scientifique. This college had been set up in 1908 as part of the Gymnase de la Cité. There he met Georges de Rham who arrived at the school one year later. Karamata graduated from the Gymnase Scientifique in 1920 and, World War I now being over, returned to his homeland which had undergone major political changes.

The end of World War I in 1918 saw the collapse of the Austro-Hungary Empire and the creation of the independent country Yugoslavia with Belgrade as its capital. Karamata's father, Stevan Karamata, moved to Belgrade where two banks were merging to become the Jadransko-Podunavska Banka. This was formally established on 11 May 1924 with Stevan Karamata as its Managing Director. Stevan Karamata had seen the increasingly attractive economic and financial potential of Belgrade and had advised his son Jovan Karamata to study civil engineering. Following this advice, in 1920 Jovan had enrolled in the Department of Civil Engineering at the Faculty of Technical Sciences of the University of Belgrade. In 1922, when he had completed two years of this course, he was advised by one of his professors, Bogdan Gavrilovic (1864-1947), to transfer from engineering to mathematics. Gavrilovic taught mathematics, physics and philosophy at the University of Belgrade but his main interests were in pure mathematics. He had written important texts including

*Analytical Geometry*,

*Theory of Determinants*, and

*Linear Algebra*. Following his advice, Karamata transferred to study mathematics in the Faculty of Philosophy at the University of Belgrade.

The leading mathematician at Belgrade at this time was Mihailo Petrović and he played an important role in Karamata's university education. Aleksandar Nikolic writes [5]:-

Karamata did not mind much for formal school knowledge, and even as a student he aspired to do independent research work. The first teacher, an example and role model in scientific work, a man he respected all his life, was Mihailo Petrović. According to the Serbian mathematician, academician Miodrag Tomic, he was the one who passed on to him the great and sincere love of science, the wide outlook, the desire to do scientific work, ideas free from formal restraints, and mathematical directions one should take in search of results.Karamata graduated in 1925 and he was appointed as a temporary assistant of Mihailo Petrović. He was already undertaking research and he explained how his found the right area [10]:-

I wanted to improve my knowledge of the foundations of the theory of functions. This is the reason why I first started to study the theory of series, but when I entered into this theory, I found so many old and new results that I remained there.He presented his first work on series to the Serbian Academy of Sciences and Arts in a lecture he gave on 14 December 1925; this would be published by the Academy. It also formed part of his doctoral thesis

*O jednoj vrsti granica slicnih odredenim integralima*Ⓣ on which he was examined on 9 March 1926. On 22 March 1926 he sat his Ph.D. examination before the same examiners who had conducted his thesis oral. He passed the examination and graduated as doctor of philosophy. A few days later he submitted

*Sur Certaines Limites Rattachées aux Intégrales de Stieltjes*Ⓣ to the Paris Academy of Sciences. Jacques Hadamard wrote (see for example [5]:-

M Karamata's astute ideas partly coincide with some of those presented by Paul Lévy. On the other hand, the two very different methods in which these two authors came to the solution, demonstrates the productivity of a new kind of thinking.Before continuing his career, he had to undertake military service. He served for one year from April 1926 to April 1927. He took the examination to become an officer at the military school but failed with the grade "incompetent" having been asked to list all the lakes in the Kingdom of Serbs, Croats and Slovenes but he missed one out!

After his military service was complete, he was awarded a Rockefeller Scholarship which enabled him to spent the period December 1927 to September 1928 in Paris. He made many friends during this visit, for example he had a lasting friendship with both André Lichnerowicz and Paul Montel. While in Paris, he had ideas for two papers which would be highly significant. Let us stress that he already had written several papers, in fact eleven of his papers were published between 1926 and 1929, five in Serbian and six in French. The authors of [1] write:-

Two works from this period have a particular place and importance and have made Karamata a mathematician of world reputation. Both papers were published in 1930, the first one in the renowned German journal "Mathematische Zeitschrift", the paper 'Über die Hardy-Littlewoodsche Umkehrungen des Abelschen Stätigkeitssatzes', while the second one in the little-known Romanian journal "Mathematica" (Cluj), 'Sur une mode de croissancerégulière des fonctions'. The first paper, which consisted only of two pages, provides a new, concise and elegant proof of the Littlewood's theorem in the theory of summability of infinity series.The impact of this paper can be seen from G H Hardy's reaction reported by Vojislav Maric via Edward Copson [4]:-

When visiting the St Andrews University in Scotland I was introduced to E T Copson (the author of widely used textbook on the theory of complex functions). He told me: "I have known only one Yugoslavian mathematician - Karamata. When cooperating with Hardy at Oxford I found him one day nervously pacing up and down his office. Not responding to greetings he abruptly said: 'I just received a letter from a young man in Belgrade who claims he can prove the Hardy-Littlewood theorem in just two pages! This is simply impossible!'"G H Hardy was wrong, it was not impossible, for Karamata had indeed proved this important theorem in two pages and his method would provide insightful and give a new method which led to further advances. In 1979

*Mathematische Zeitschrift*celebrated 60 years of publishing mathematics and listed what they considered the 50 most important papers published in these 60 years. Karamata's 2-page 1930 paper was one of the 50.

We mentioned two highly significant papers above, so let us quote from the authors of [1] about the second:-

The second paper established the foundations for the theory of regularly varying functions, which will later find numerous applications in probability theory, number theory, differential equations, complex analysis, and even mathematical analysis of cosmological parameters.Returning to Belgrade, Karamata began a period of remarkably intense work. The authors of [11] write:-

The most significant period of Karamata's work was from 1927 to 1937, or perhaps between 1929 and 1933. He was obsessed with mathematics then. In order to be free, in order to be able to think solely and uninterruptedly about mathematics, he scheduled all his classes (he was an assistant professor then) on one day, from eight o'clock in the morning until eight in the evening. And those courses were: Introduction to Analysis, Elementary Algebra, Algebra , Foundations of Geometry, Descriptive Geometry, Introduction to the Theory of Functions, and a Seminar. Although in ill health, he spoke without stopping, in a classroom where students and subjects followed each other in constant succession. One can see from this itself that when he worked he didn't let up. He made many innovations in the teaching process, which hadn't changed in decades. A great many of the students, accustomed to formality in the teaching process, were perplexed and dissatisfied. In the seminars, instead of the literature, textbooks and monographs, he used research papers and recent ones at that.Karamata was promoted to assistant professor in 1930 and in the following year he married the lawyer Emilija Nikolajevic (1906-1959), the daughter of the Belgrade judge Borislav Nikolajevic. Jovan and Emilija Karamata had four children: two sons, Vladimir Karamata (born in 1934, who became an architect in Geneva) and Dimitri Karamata (born in 1935, who became a professor at the Medical Faculty of the University of Lausanne), and twin daughters, Maria (1938-1940) and Katarina (1938-2005, who became a biologist in Geneva). Emilija was fluent in French and German and she assisted Karamata by typing his papers and taking the role of secretary in his correspondence with other mathematicians.

Karamata's mathematical output was very large with around 120 papers, 15 books (both monographs and textbooks) and 7 works about teaching. In addition to the fame from the Hardy-Littlewood theorem, he gained a world reputation for his work on slowly varying functions and his work on Tauberian theory. Tomic [1] writes:-

Karamata had the most effective period of scientific and teaching work during 1929-1939, before World War II started. These ten years were marked not only by the appearance of his most important work, but also by attempts to introduce new directions in teaching. This has stimulated the awakening of mathematics in Yugoslavia which, until then, followed outdated fixed paths. Apart from greatest efforts in scientific work, Karamata also had other activities. He visited many universities in Europe and presented many lectures on his investigations. He participated in almost all of the largest scientific meetings and conferences between the two world wars, he met many scientists with whom he not only corresponded, but also had some sort of friendly relationship - among others several famous mathematicians of that time, such as: E Landau, Knopp, I Schur, Fejér, M Riesz, P Montel, W Blaschke, ... His reviews in the 'Zentralblatt für Mathematik' and in the 'Jahrbuch über die Fortschritte der Mathematik' were noted for their criticism and conscientious comments.In 1937 he was promoted to associate professor but World War II, beginning in 1939, proved a difficult time for Karamata. He had always tried to avoid expressing any political views, concentrating on his mathematics and his family. In a collaboration with a friend, he had, however, built a textile factory which began operation just before the start of the war. He had also set up a blacksmiths' workshop behind his own house. On 6 April 1941 Germany, Italy and Hungary all invaded Yugoslavia and within a week Belgrade was occupied. The different races in Yugoslavia now reacted in different ways, too complicated to attempt to explain here. Karamata did not accept to rule of Milan Nedic's government, so was dismissed from his university position. Before the end of 1941, the German Ministry of Education offered him a position in any German university of his choice, but he refused. He returned to his hometown of Zemun, now in the occupied Independent State of Croatia, and there with skill and bribery managed to survive the war years. He was arrested several times but, with textiles and output from the blacksmith shop, he was able to use bribery to ensure his release. Although the textile factory was supposed to work to support the invaders, somehow he managed to ensure it did not. When the invaders were expelled, Karamata was again arrested in the belief he had collaborated. Again he survived, but the textile factory was nationalised and he lost almost all of his blacksmiths' shop.

When the war ended in 1945, the University of Belgrade reopened. It was, however, a strange time since there were five years of students all wanting to begin their studies at the same time. Karamata worked hard to assist these students but times were difficult without most of his former friends and with new ideas about teaching but no new books. He attempted to help remedy the situation by publishing textbooks. His book

*Theory and application of Stieltjes integral*(Serbian) (1949) was reviewed by William Feller:-

A textbook containing a careful exposition of all details and a wealth of examples and counter-examples. There are four main parts. Some 36 pages are devoted to general properties of functions of bounded variation. Next some 70 pages treat the Stieltjes integral, mean value theorems, etc. The third and main part [about 110 pages] is called "applications." Its chapter headings are, approximately: (1) Functions of sequences of numbers, (2) Applications to infinite series, (3) General summation formulas, (4) Special summation formulas, (5) Dirichlet series, (6) Behaviour of Dirichlet series on the boundary of the domain of convergence, (7) Analytic continuation to the left of the abscissa of convergence. The last section [about 95 pages] contains various notes and explanations concerning infinite series, calculus, and inequalities.The review in [1] states that Karamata's

*Complex Numbers with Applications to Elementary Geometry*(Serbian) (1950, 1960):-

... consists of two parts - 'Algebra of Complex Numbers' and 'Applications in Geometry'. In the first part, the notion of a complex number is informally introduced in relation to solving algebraic equations. Afterwards, the structure of complex numbers is based on the axiomatic method. The concepts of general algebra are mentioned, such as algebraic structures of rings - associative and commutative law, as well as the law of the neutral element and others. Using these identities, the other algebraic properties of the considered algebraic operations are formally derived. The field of complex numbers is defined as it is generally accepted today - complex numbers are ordered pairs (complexes) with common definitions of addition and multiplication operations. Trigonometric and Euler's formulas for complex numbers are also introduced, as well as power and exponential functions for complex numbers.Given Karamata's background, despite his attempts to fit in, it was inevitable that he would be considered "a reactionary bourgeois capitalist" by the Communist regime. His influence in the Department of Mathematics was minimal because he was seen as unfriendly towards the regime. He became frustrated and, despite being elected a full member of the Serbian Academy of Sciences and Arts in 1948, and promoted to full professor at the University of Belgrade in 1950, he decided to accept a full professorship at the University of Geneva in the following year [10]:-

Karamata's commitment to young talent in Belgrade didn't slacken even after he moved to the University of Geneva in 1951. On the contrary, it only intensified. During his frequent and regular visits to Belgrade he magnetically attracted young mathematics students who had never even seen him before, let alone heard him in the classroom. He would meet with them individually, always in regard to some mathematical problem, usually at the Hotel Majestic, or else frequently - in groups, at the Mathematical Institute of the Serbian Academy of Sciences. The conversation topics would be related to some specific problem) or to general developments in mathematics.Karamata's wife Emilija died in 1959 and after this he seems to have become less happy with the direction of the Department of Mathematics at Geneva. He made many trips to the United States, particularly as a visiting professor at the Mathematical Research Center in Madison, Wisconsin. There he collaborated with other mathematicians who, like him, had chosen to leave Yugoslavia. He now visited Belgrade less often only going to Yugoslavia to visit his brothers. Although aware that his health was deteriorating, he never complained and tried to avoid seeing medical doctors. He certainly did not realise how close he was to death, for he was organising a trip to the United States with plans to lecture at several universities. He became seriously ill and was admitted to hospital on 13 July 1967. He died one month later from cancer. Both Karamata and his wife were heavy smokers and subsequently they both died prematurely from cancer. Following his wishes, his ashes were laid to rest in Zemun.

### References (show)

- D Brankovic and Z Mijajloviv, Jovan Karamata and his digitized works,
*NCD Review*(Bulgarian)**35**(2019), 28-38. - M Dabizic, The Zemun House of Jovan Karamata with a Memorial Plaque (Serbian),
*Monumental Heritage*(Serbian)**11**(2010), 103-107.

https://www.uni-miskolc.hu/~matsefi/HMTM_2020/papers/HMTM_2020_Nikolic_Karamata.docx - Jovan Karamata (1902-1967),
*Mathematical Institute of the Serbian Academy of Sciences and Arts National Institute of the Republic of Serbia*.

http://www.mi.sanu.ac.rs/novi_sajt/aboutus/history/distiguished_mathematicians/karamata.php - V Maric, Jovan Karamata (1902-1967),
*Mathematics Journal*(Bulgarian)**54**(2002), 45-51. - A M Nikolic, The intellectual biography of a mathematician,
*University of Miskolc*(2020).

https://www.uni-miskolc.hu/~matsefi/HMTM_2020/papers/HMTM_2020_Nikolic_Karamata.docx - A M Nikolic, Jovan Karamata (1902-1967),
*University of Miskolc*(2020).

https://www.uni-miskolc.hu/~matsefi/HMTM_2020/annexes/HMTM_2020_Nikolic_JovanKaramata.pdf - A M Nikolic, Jovan Karamata (1902-1967),
*Novi Sad J. Math.***32**(1) (2002), 1-5. - A M Nikolic, About two famous results of Jovan Karamata,
*Archives Internationales d'Histoires des Sciences***141**(48) (1998), 354-373. - A M Nikolic, Jovan Karamata (1902-1967), Lives and work of the Serbian scientists (Serbian), Biographies and bibliographies
**5**(*Serbian Academy of Sciences and Arts*, Belgrade, 1999), 235-283. - M Tomic, Jovan Karamata (1902-1967),
*Bulletin de l'Acad. Serbe des Sci. et des Arts***122**(26) (2001), 1-29.

https://www.emis.de/journals/BSANU/26/r2001_1.pdf - M Tomic and S Aljancié, Remembering Jovan Karamata,
*Publications de l'Institut Mathématique***48**(62) (1990), 1-6. - M Tomic, Academician Jovan Karamata, on occasion of his death (Serbian),
*Serbian Academy of Sciences and Arts*173 (Belgrade, 1968), 1-27. - M Tomic, Jovan Karamata (1902-1967),
*L'Enseignement Mathématique***15**(1969), 1-20.

### Additional Resources (show)

Other pages about Jovan Karamata:

Other websites about Jovan Karamata:

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update March 2022

Last Update March 2022