ššššššš 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.
Last updated:š 23.08.99