On Tuesday, 7 November, at 16:20 in room 5234 NSU the next meeting of the seminar on geometric analysis will be held.
Speaker: Vladimir Nikolaevich Dubinin, Doctor of Physics and Mathematics, Corresponding Member. RAS
Title of the report: Boundary distortion and Schwartz derivative of a univalent function in a circular ring.
Annotation:
A lot of work has been devoted to questions of the boundary behavior of conformal mappings. This lecture presents proofs of new distortion theorems for holomorphic univalent and bounded functions in a circular ring that preserve one of its boundary components. In particular, inequalities are established that include the Schwarz derivative at the boundary point of the ring. Studies of various classes of holomorphic functions in a ring, carried out before 1966, are quite fully presented in the “addendum” to G.M. Goluzin’s book. The development of the methods used in this case is also reflected in the review by G.V. Kuzmina. In modern literature, as a rule, functions defined in simply connected domains are considered, and the more complex case of a ring has been studied to a lesser extent. Let us note the significant results of A.Yu. Solynin in the study of holomorphic functions in a doubly connected domain. Our lecture complements the research begun in the work, with the aim of showing the effectiveness of using the capacitive approach and symmetrization to obtain distortion theorems for functions univalent in a ring. The extremal function will be the n-fold symmetric Gretsch function, mapping a circular ring onto another circular ring with n symmetric radial cuts. Using only the simplest properties of conformal capacity, we prove a fairly general theorem on the product of powers of moduli of derivatives of a univalent function at boundary points of a ring. By standard deductions, this result entails a distortion theorem for holomorphic mappings of a circle into itself. The latter contains some inequalities from the works. Drawing on our earlier result on the product of the internal radii of mutually non-overlapping regions in a circular ring, we establish an upper bound for the geometric mean moduli of the derivatives at the symmetric boundary points of the ring. The lower bound for the arithmetic means of such modules is obtained using symmetrization. In conclusion, we present proofs of Schwarzian estimates at the boundary points of a ring, which are a far-reaching continuation of the research initiated by the famous Burns-Krantz theorem on the rigidity of holomorphic mappings of a circle into itself. The lecture is based on materials from a recent article written during research work at the Mathematical Center in Akademgorodok.
