On the Riemann–Hilbert problem for analytic functions in circular domains

TitleOn the Riemann–Hilbert problem for analytic functions in circular domains
Publication TypeJournal Article
Year of Publication2016
AuthorsYefimushkin, AS, Ryazanov, VI
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published2/2016

The existence of single-valued analytic solutions in a unit disk and multivalent analytic solutions in domains bounded by a finite collection of circles is proved for the Riemann–Hilbert problem with coefficients of sigma finite variation and with boundary data that are measurable with respect to the logarithmic capacity. It is shown that these spaces of solutions have the infinite dimension.

Keywordsanalytic functions, circular domains, Riemann–Hilbert problem
  1. Vekua I. N. Obobschennyie analiticheskie funktsii, Moscow: Fizmatgiz, 1959 (in Russian).
  2. Efimushkin A. S., Ryazanov V. I. Ukr. mat. vestnik, 2015, 12, No 2: 190–209 (in Russian).
  3. Karleson L. Izbrannyie problemyi teorii isklyuchitelnyih mnozhestv, Moscow: Mir, 1971 (in Russian).
  4. Nevanlinna R. Odnoznachnyie analiticheskie funktsii, Moscow: OGIZ, 1941 (in Russian).
  5. Nosiro K. Predelnyie mnozhestva, Moscow: Izd-vo Inostr. lit., 1963 (in Russian).
  6. Fékete M. Math. Z., 1923, 17: 228–249.
  7. Goluzin G. M. Geometricheskaya teoriya funktsiy kompleksnogo peremennogo, Moscow: Nauka, 1966 (in Russian).
  8. Twomey J. B. Irish Math. Soc. Bulletin., 2006, 58: 81–91. doi: https://doi.org/10.1515/math-2015-0034
  9. Kusis P. Vvedenie v teoriyu prostranstv Hp, Moscow: Mir, 1984 (in Russian).
  10. Ryazanov V. Open Math., 2015, 13, No 1: 348–350.