**Walter Hayman**'s parents were Franz Karl Abraham Samuel Haymann (the double 'n' is not a misprint) and Ruth Therese Hensel. Ruth Hensel was the daughter of the famous mathematician Kurt Hensel and his wife Gertrud Hahn. In 1933, Kurt Martin Hahn, Gertrud's niece, was forced to leave Germany because he was Jewish and he went to Scotland where in the following year he founded the internationally famous Gordonstoun School near Elgin in the north east of Scotland. Walter is therefore related to famous mathematicians, famous educationalists, famous philosophers, and to the brilliant composers Fanny and Felix Mendelssohn (see Kurt Hensel's biography). Franz Haymann (1874-1947) was Professor of Law at the University of Cologne when Walter was born. However, being Jewish, he was forced to retire from his chair in September 1935 by the Nazi government. He emigrated to Britain in 1938, and it will come as no surprise to learn that his son Walter was sent to Scotland to be educated at Gordonstoun School.

After graduating from Gordonstoun School, Hayman studied at St John's College, Cambridge. There he was particularly influenced by his lecturers Mary Cartwright and J E Littlewood, and he went on to undertake research supervised by Mary Cartwright. In 1947 he was appointed as a lecturer in King's College, Newcastle and in the same year he became a fellow of St John's College, Cambridge. Later in 1947 he was appointed as a Lecturer in Mathematics at the University of Exeter. His first mathematics paper *Some remarks on Schottky's theorem* was published in the *Proceedings of the Cambridge Philosophical Society* in 1947. It calculates numerical estimates in Schottky's theorem, based on a detailed discussion of the elliptic modular function by geometric methods. This paper was followed by *Remarks on Ahlfors' distortion theorem* (1948) and *Some inequalities in the theory of functions* (1948). Lars Ahlfors, discussing the second of these papers, explains that "The results seem very accurate, but their formulation is extremely complicated." He also talks of Hayman's "skillful handling of technical points". At Cambridge, Hayman was awarded the 1^{st} Smith's Prize in 1948 and in the following year he shared the Adams Prize with John Charles Burkill, Subrahmanyan Chandrasekhar and John MacNaughton Whittaker.

Hayman's fellowship at St John's College, Cambridge continued until 1950 and he was a Visiting Lecturer at Brown University in Providence, Rhode Island during 1949-50. Hayman spent the summer of 1950 at Stanford University in California. While there he wrote the Technical Report for a Navy Contract on *Symmetrization in the theory of functions*. W Seidel writes:-

The author investigates a large number of applications to the theory of functions of two methods of symmetrization: the Steiner method, recently treated by Pólya and Szegö, and circular symmetrization, introduced by Pólya.

At the University of Exeter, Hayman was promoted to Reader in Mathematics in 1953. His work was attracting much interest and he was an Invited Speaker at the International Congress of Mathematicians held in Amsterdam in September 1954. In 1955 he gave a proof of the asymptotic Bieberbach conjecture and was awarded the Junior Berwick Prize by the London Mathematical Society. Again he spent the summer of 1955 visiting Stanford University. He left Exeter in 1956 when he was appointed as the first Professor of Pure Mathematics at Imperial College, London [5]:-

For over30years there he ran a school of renown in Complex Analysis, attracting mathematicians from all over the world.

Also in 1956 Hayman was elected a fellow of the Royal Society. However, before we continue to describe his career we should return for a moment to 1947 for that was the year in which he married Margaret Riley Crann, the daughter of Thomas Crann of New Earswick, York. Margaret had studied mathematics at Cambridge and been awarded an M.A. She became a mathematics teacher at the Putney High School for Girls and has become famous for her work with the most mathematically gifted pupils. In 1966 Walter Hayman visited Moscow and when he returned, together with his wife, he founded the British Mathematical Olympiad. The Haymans set and marked the questions for the British Olympiad Competition. Their efforts [3]:-

... led to the United Kingdom becoming one of the first western countries to send a team to the International Mathematical Olympiad[in1967] (which was, in its earliest days, the preserve of the communist states). [Margaret Hayman]waged a long battle for sponsorship funding for the British team, and worked hard to foster appreciation of the potential value of mathematics competitions within the Council and the membership of the Mathematical Association.

Walter and Margaret had three daughters. One of these was Anne Carolyn Hayman, born on 23 April 1951, who was educated at Putney High School for Girls and then went on to obtain a B.A. in Classics and Philosophy. She received an OBE in 2003 and, since 2004, has been Chief Executive of Peace Direct.

While at Imperial College, Hayman published many deep research articles and also superb monographs skilfully written so that only undergraduate mathematics is required as a prerequisite. In 1958 he published the 151-page *Multivalent functions*. W C Royster begins a review as follows:-

This book deals with that part of the theory of univalent and multivalent functions which is closest to the researches of the author. The majority of the results obtained throughout the book are of the classical inequality type. However, it is not to be inferred that only elementary methods are used in obtaining these results. For those with an introductory course in complex variables this book is self-contained, which is remarkable considering the length of the tract and the level reached.

Hayman's next monograph was *Meromorphic functions* (1964). W H J Fuchs writes:-

This attractively written monograph gives a rather complete account of the Nevanlinna theory of meromorphic functions. It also treats the Ahlfors theory of covering surfaces and the theory of invariant normal families. The book can be understood by a reader familiar with basic function theory and with the elements of Lebesgue theory. A very nice feature of the book is the way in which routine discussions of standard results are interspersed with non-standard applications.

The text *Research problems in function theory* published in 1967 was somewhat different in nature. Ch Pommerenke writes about the 56-page booklet:-

This little book originated in a conference on classical function theory held in London in1964. It contains more than140open problems from the theory of functions of one complex variable and related subjects such as subharmonic functions. Most problems are connected with quantitative questions, for instance about the growth of a function and about estimates for some interesting constants. A few of the problems are classical and some were known before, but many of them are new. The background of most of the problems is sketched and the present state of knowledge indicated. ... This booklet is highly recommended to all interested in function theory. Not only does it give a wealth of research problems but also some impression of the present state of a large part of function theory.

Of all the books which Hayman wrote during his thirty years at Imperial College, the most significant was almost certainly the first volume of *Subharmonic functions* (1976). The project to write the text was a joint one by Hayman and Patrick Brendan Kennedy, but sadly Kennedy died in 1967. Hayman continued working on the book on his own but still published it under the joint authorship:-

This book gives a clear and attractive account of the theory of subharmonic functions in domains of Euclidean spaces R(^{m}m ≥2)and illuminates the connections and analogies with classical complex variable theory particularly well.

Hayman promised a second volume on the theory of subharmonic functions in the plane and he did keep his promise but it took 13 years to complete. The second volume, published in 1989, was:-

... welcomed as the definitive work on some of the most actively pursued research topics in the theory of subharmonic functions.

Perhaps the fact that during these thirteen years he was dean of the Royal College of Science from 1978 to 1981, and vice-president of the London Mathematical Society from 1982 to 1984 is relevant here. He retired from his chair at Imperial College in 1985 and was appointed as Professor of Pure Mathematics at the University of York, this being a part-time position. He held this post until he reached the age of 67 in 1993. Then, two years later, he became a Senior Research Fellow back at Imperial College in London, a position he continues to hold. Margaret Hayman died in 1994 and in the following year he married the mathematician Dr Waficka al-Katifi who had published *On the Asymptotic Values and Paths of Certain Integral and Meromorphic Functions* in the *Proceedings* of the London Mathematical Society in 1966. She died in 2001.

One further important text by Hayman that we should mention is the second edition of *Multivalent functions* published in 1994. W C Royster, who reviewed the first edition 35 years earlier, also reviewed the second edition:-

The first edition of this book dealt with the growth of univalent and multivalent functions to obtain bounds for the modulus and coefficients and related quantities. After35years of rapid development in this area the author expands the first edition to include primarily new results that are more closely related to topics in the first edition. ... the material is developed in a very interesting and structured manner. The addition of examples and exercises adds greatly to its utility for graduate students and researchers in the theory of multivalent and univalent functions.

We have already mentioned some of the honours that Hayman has received. In addition to the invitation to lecture at the International Congress of Mathematicians held in Amsterdam, which we mentioned above, he was also invited to lecture at the International Congress of Mathematicians held in Nice in 1970. He has been invited to address the British Mathematical Colloquium on four occasions, being one of a very small number of mathematicians to have that many invitations. He gave the lectures *The growth of integral functions* at Greenwich in 1952, *The modern approach to Riemann surfaces* in St Andrews in 1956, *Some Tauberian theorems for multivalent functions* in York in 1970, and *Value distribution of functions meromorphic in an angle* in Sheffield in 1980. He was awarded the Senior Berwick Prize by the London Mathematical Society in 1964, and the De Morgan Medal by the Society in 1995. The citation for this Medal reads [1]:-

Professor Hayman is awarded the De Morgan Medal for his contributions to complex analysis and potential theory. Following the tradition established by Hardy and Littlewood, he became the undisputed leading international authority in classical complex analysis, through work contained in more than one hundred research papers and several monographs. His natural creative talent for solving problems led in particular to very deep results in the theory of functions on the unit disc, integral and Schlicht functions, and subharmonic functions in several variables. As an international figure, Professor Hayman has influenced researchers of all nations and of all ages during his periods at both Imperial College and the University of York. He has also been much involved with the role of mathematics and science in society, and in supporting the human rights of individual mathematicians. Few have been as dedicated to their chosen area of mathematics as Walter Hayman, or promoted it in the world in such a dedicated fashion.

He has been elected to several academies. We mentioned his election to the Royal Society in 1956 but we should also record that he served on the Council of that Society in 1962-63. He was elected to the Finnish Academy of Science and Letters (1978), the Bavarian Academy (1982), and the Accademia dei Lincei of Rome (1985). He has received honorary degrees from the University of Exeter (1981), the University of Birmingham (1985), Uppsala University, Sweden (1992), Giessen University, Germany (1992), and the National University of Ireland, Dublin (1997). Another honour we must mention is the special 2008 volume of *Computational Methods and Function Theory* dedicated to Hayman on the occasion of his 80^{th} birthday. We quote from the Preface [5]:-

Walter has written over200papers touching and enriching all areas of complex analysis. He is best known, perhaps for his work on the minimum modulus of large entire functions, for his proof of the asymptotic Bieberbach conjecture and for the so-called "Hayman Alternative" in value distribution theory. In addition he has written three books, "Multivalent Functions", "Meromorphic Functions" and "Subharmonic functions Vols.1and2"(the first volume jointly with P B Kennedy)which have been extremely influential. He now lives in the country, in Bisley, Gloucestershire but still maintains an active scientific and academic life in London. The present volume is dedicated to Walter by his many students, friends and admirers.

Finally we note that Hayman gives "music, travel, television" as his hobbies.

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

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