Leningrad school of the geometric theory of functions


The term "geometric theory of functions" denotes the part of functional analysis devoted to estimations of different magnitudes related to a conformal mapping of one region onto another. This area is very rich in results, both applied and purely theoretic. Theorems of the geometric theory of functions are remarkable by their particular elegance and simplicity of formulations (often delusive).
        The research in the geometric theory of functions in Leningrad was initiated in the late twenties by Gennady Mikhailovich Goluzin, a pupil of Vladimir Ivanovich Smirnov. Soon he became one of the most authoritative and bright representatives of mathematicians working in this area. Since we cannot cite here all numerous and profound results of G.M.Goluzin, we mention only the famous "Goluzin variational principle" and his remarkable monograph "Geometric theory of functions of complex variables".
        A great contribution to the theory was made by Goluzin's pupils - Yu.E.Alenitsyn, N.A.Lebedev, I.M.Milin. The Leningrad school of the geometric theory was formed in the late forties. Professor Nikolai Andreevich Lebedev headed the Leningrad seminar of the geometric theory of functions of complex variables for more than thirty years (at present this seminar continues its work at St.Petersburg Department of Steklov Institute of Mathematics, headed by G.V.Kuzmina). The participation of N.A.Lebedev enlivened the seminar. Whatever was the subject of a talk, he always was able to find a sensible idea in it and to see the ways of further research. Lebedev's pupils G.V.Kuzmina, N.A.Shirokov and others obtained interesting results connecting the geometric theory with other parts of analysis, in particular, with the approximation theory. These relations are reflected in a well known monograph "Constructive theory of functions of complex variables" by V.I.Smirnov and N.A.Lebedev and in the monograph "Area principle in the theory of univalent functions" by N.A.Lebedev. In 1984 an event took place that disturbed the mathematical community. An American mathematician De Brange solved the "coefficients problem" which had been open for more than half a century in spite of efforts of many researchers, including outstanding ones. An essential element of the work of De Brange was a beautiful inequality obtained by Leningrad mathematicians Lebedev and Milin. The success of De Brange was to some extent prepared by some works of I.M.Milin. The original proof of De Brange was very ponderous. He brought it to Leningrad where it was laboriously checked with an active participation of G.V.Kuzmina, I.M.Milin and E.G.Emel'yanov. They managed to free the proof from unnecessary and ponderous details and to make it sufficiently clear.
        New bright results on the geometric theory were obtained by N.G.Makarov (at the time he worked at the Chair of Mathematical Analysis). Using some probabilistic methods he obtained an exhaustive solution of difficult problems on the boundary behaviour of a conformal mapping of the circle onto a Jordan region. The works of N.G.Makarov were awarded the International Salem Prize in 1986.

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Last  updated: 23.08.99 .