Siberian Mathematical Journal

, Volume 45, Issue 4, pp 606–617 | Cite as

On the Bohr Radius for Two Classes of Holomorphic Functions

  • L. Aizenberg
  • A. Vidras


Using some multidimensional analogs of the inequalities of E. Landau and F. Wiener for the Taylor coefficients of special classes of holomorphic functions on Reinhardt domains we obtain some estimates for the Bohr radius.

Bohr radius hypercone 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bohr H., "A theorem concerning power series," Proc. London Math. Soc., 13, 1–5 (1914).Google Scholar
  2. 2.
    Landau E. and Gaier D., Darstellung und Bergrundung eininger neurer Ergebnisse der Functionentheorie, Springer-Verlag, Berlin etc. (1986).Google Scholar
  3. 3.
    Sidon S., "Uber einen Satz von Herrn Bohr," Math. Z., 26, 731–732 (1927).Google Scholar
  4. 4.
    Tomic M., "Sur une theorem de H. Bohr," Math. Scand., 11, 103–106 (1962).Google Scholar
  5. 5.
    Dineen S. and Timoney R. M., "Absolute bases, tensor products and a theorem of Bohr," StudiaMath.,94,No.2, 227–234 (1989).Google Scholar
  6. 6.
    Dineen S. and Timoney R. M., "On a problem of H. Bohr," Bull. Soc. Roy. Sci. Liege, 60, No. 6, 401404 (1991).Google Scholar
  7. 7.
    Boas H. P. and Khavinson D., "Bohr's power series theorem in several variables," Proc. Amer. Math.Soc.,125, 2975–2979 (1997).CrossRefGoogle Scholar
  8. 8.
    Aizenberg L., "Multidimensional analogues of Bohr's theorem on power series," Proc. Amer. Math. Soc., 128, 1147–1155 (2000).CrossRefGoogle Scholar
  9. 9.
    Boas H. P., "Majorant series," J. Korean Math. Soc., 37, 321–337 (2000).Google Scholar
  10. 10.
    Aizenberg L., Aytuna A., and Djakov P., "An abstract approach to Bohr's phenomenon," Proc. Amer. Math. Soc., 128, 2611–2619 (2000).CrossRefGoogle Scholar
  11. 11.
    Aizenberg L., Aytuna A., and Djakov P., "Generalization of Bohr's theorem for bases in spaces of holomorphic functions of several complex variables," J. Math. Anal. Appl., 258, 428–447 (2001).CrossRefGoogle Scholar
  12. 12.
    Aizenberg L. and Tarkhanov N., "A Bohr phenomenon for elliptic equations,"Proc. London Math. Soc., 82, 385–401 (2001).CrossRefGoogle Scholar
  13. 13.
    Aizenberg L., "Bohr theorem," in: Encyclopedia of Mathematics, Supplement II (ed. M. Hazewinkel), Kluwer, Dordrecht, 2000, pp. 76–78.Google Scholar
  14. 14.
    Aizenberg L., "Generalization of Carathéodory's inequality and the Bohr radius for multidimensional power series," Complex Variables Theory Appl. (to appear).Google Scholar
  15. 15.
    Aizenberg L., Liflyand E., and Vidras A., "Multidimensional analogue of van der Corput-Visser inequality and its application to the estimation of the Bohr radius," Ann. Polon. Math., 80, 47–54 (2003).Google Scholar
  16. 16.
    Aizenberg L. A., Grossman I. B., and Korobeinik Yu. F., "Some remarks on Bohr radius for power series," Izv. Vuzov, No. 10, 3–10 (2002).Google Scholar
  17. 17.
    Djakov P. B. and Ramanujan M. S., "A remark on Bohr's theorem and its generalizations," J. Anal., 8, 65–77 (2000).Google Scholar
  18. 18.
    Glazman I. M. and Lyubich Yu. I., Finite-Dimensional Linear Analysis [in Russian], Nauka, Moscow (1969).Google Scholar
  19. 19.
    Dixon P. G., "Banach algebras satisfying the non-unital von Neumann inequalities," Bull. London Math. Soc., 27, 359–362 (1995).Google Scholar
  20. 20.
    Nikolskii N. K., Operators, Functions and Systems: An Easy Reading. Vol. 1 and 2, Amer. Math. Soc., Providence RI (2002).Google Scholar
  21. 21.
    Paulsen V. I., Popescu G., and Singh D., "On Bohr's inequality," Proc. London Math. Soc. (3), 85, 493–515 (2002).CrossRefGoogle Scholar
  22. 22.
    Defant A., Garcia D., and Maestre M., "Bohr's power series theorem and local Banach space theory," J. Reine Angew. Math., 557, 173–197 (2003).Google Scholar
  23. 23.
    Aizenberg L. A. and Mityagin B. S., "The spaces of functions analytic in multiply circular domains," Sibirsk. Mat. Zh., 1, No. 2, 153–170 (1960).Google Scholar
  24. 24.
    Aizenberg L., "Integral representations of functions holomorphic in n-circular domains (continuation of Szego kernels)," Mat. Sb. (N.S.), 65(107), 104–143 (1964).Google Scholar
  25. 25.
    Aizenberg L. and Yuzhakov A., Integral Representations and Residues in Multidimensional Complex Analysis, Amer. Math. Soc., Providence RI (1983).Google Scholar
  26. 26.
    Prudnikov A. P., Brychkov Yu. A., and Marychev O. I., Integrals and Series. Vol. 1: Elementary Functions, Gordon & Breach Science Publishers, New York etc. (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • L. Aizenberg
    • 1
  • A. Vidras
    • 2
  1. 1.Bar-Ilan UniversityRamat Gan
  2. 2.University of CyprusNicosia

Personalised recommendations