Advertisement

Complex Analysis and Operator Theory

, Volume 6, Issue 5, pp 1025–1036 | Cite as

Landau–Bloch Constants for Functions in α-Bloch Spaces and Hardy Spaces

  • Sh. Chen
  • S. PonnusamyEmail author
  • X. Wang
Article

Abstract

In this paper, we obtain a sharp distortion theorem for a class of functions in α-Bloch spaces, and as an application of it, we establish the corresponding Landau’s theorem. These results generalize the corresponding results of Bonk, Minda and Yanagihara, and Liu, respectively. We also prove the existence of Landau–Bloch constant for a class of functions in Hardy spaces and the obtained result is a generalization of the corresponding result of Chen and Gauthier.

Keywords

Landau–Bloch constant Holomorphic function α-Bloch space Hardy space 

Mathematics Subject Classification (2000)

Primary 32A10 Secondary 32A17 32A18 32A35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlfors L.V.: An extension of Schwarz’s lemma. Trans. Am. Math. Soc. 43, 359–364 (1938)MathSciNetGoogle Scholar
  2. 2.
    Bochner S.: Bloch’s theorem for real variables. Bull. Am. Math. Soc. 52, 715–719 (1946)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bonk M.: On Bloch’s constant. Proc. Am. Math. Soc. 378, 889–894 (1990)MathSciNetGoogle Scholar
  4. 4.
    Bonk M., Minda D., Yanagihara H.: Distortion theorems for Bloch functions. Pac. J. Math. 179, 241–262 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chen H., Gauthier P.M.: Bloch constants in several variables. Trans. Am. Math. Soc. 353, 1371–1386 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chen H., Shiba M.: On the locally univalent Bloch constant. J. Analyse Math. 94, 159–170 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Duren P.: Theory of H p Spaces. Academic press, New York (1970)zbMATHGoogle Scholar
  8. 8.
    Fitzgerald C.H., Gong S.: The Bloch theorem in several complex variables. J. Geom. Anal. 4, 35–58 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Graham I., Varolin D.: Bloch constants in one and several variables. Pac. J. Math. 174, 347–357 (1996)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hahn K.T.: Higher dimensional generalizations of the Bloch constant and their lower bounds. Trans. Am. Math. Soc. 179, 263–274 (1973)zbMATHCrossRefGoogle Scholar
  11. 11.
    Harris L.A.: On the size of balls covered by analytic transformations. Monatsh. Math. 83, 9–23 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Landau E.: Über die Bloch’sche konstante und zwei verwandte weltkonstanten. Math. Z. 30, 608–634 (1929)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Li S.X.: Riemann-Stieltjes operators between α-Bloch spaces and besov spaces. Math. Nachr. 282, 899–911 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Li S.X., Wulan H.: Characterizations of α-Bloch spaces on the unit ball. J. Math. Anal. Appl. 337, 880–887 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu X.Y.: Bloch functions of several complex variables. Pac. J. Math. 152, 347–363 (1992)zbMATHGoogle Scholar
  16. 16.
    Liu X.Y., Minda C.D.: Distortion theorems for Bloch functions. Trans. Am. Math. Soc. 333, 325–338 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Minda D.: Bloch constants. J. Analyse Math. 41, 54–84 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Minda, D.: Marden constants for Bloch and normal functions. J. Analyse Math. 42, 117–127 (1982/83)Google Scholar
  19. 19.
    Minda, D.: The Bloch and Marden constants. Comput. Methods Funct. Theory 1435, 131–142. Lecture Notes in Math., Springer, Berlin (1990)Google Scholar
  20. 20.
    Pommerenke Ch.: On Bloch functions. J. Lond. Math. Soc. 2, 689–695 (1970)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Takahashi S.: Univalent functions in several complex variables. Ann. Math. 53, 464–471 (1951)zbMATHCrossRefGoogle Scholar
  22. 22.
    Wu H.: Normal families of holomorphic functions. Acta Math. 119, 193–233 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Zhang X., Xiao J.: Weighted composition operator between μ-Bloch spaces on the unit ball. Sci. China Ser. A Math. 48, 1349–1368 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Zhu K.: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 23(3), 1143–1177 (1993)zbMATHCrossRefGoogle Scholar
  25. 25.
    Zhu K.: Operator Theory in Function Spaces. Marcel Dekker, New York (1990)zbMATHGoogle Scholar
  26. 26.
    Zhu K.: Spaces of Holomorphic Functions in the Unit Ball. Springer, New York (2005)zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education)Hunan Normal UniversityChangshaPeople’s Republic of China
  2. 2.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

Personalised recommendations