Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An extremal function for the Chang-Marshall inequality over the Beurling functions

Author: Valentin V. Andreev
Journal: Proc. Amer. Math. Soc. 133 (2005), 2069-2076
MSC (2000): Primary 30H05; Secondary 30A10, 30D99, 49K99
Published electronically: January 31, 2005
MathSciNet review: 2137873
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: S.-Y. A. Chang and D. E. Marshall showed that the functional $\Lambda(f) =(1/2\pi) \int _0^{2\pi }\exp \{ \vert f(e^{i\theta})\vert^2\}d\theta$ is bounded on the unit ball $ \mathcal{B} $ of the space $ \mathcal{D}$ of analytic functions in the unit disk with $f(0)=0$ and Dirichlet integral not exceeding one. Andreev and Matheson conjectured that the identity function $f(z)=z$ is a global maximum on $ \mathcal{B}$ for the functional $\Lambda $. We prove that $\Lambda $ attains its maximum at $f(z)=z$ over a subset of $ \mathcal{B}$ determined by kernel functions, which provides a positive answer to a conjecture of Cima and Matheson.

References [Enhancements On Off] (What's this?)

  • 1. Valentin V. Andreev, and Alec Matheson, Extremal functions and the Chang-Marshall inequality, Pacific J. Math. 162 (1994), no. 2, 233-246. MR 1251899 (95f:30051)
  • 2. Albert Baernstein, II, Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139-169. MR 0417406 (54:5456)
  • 3. S.-Y. A. Chang, and D. E. Marshall, On a sharp inequality concerning the Dirichlet integral, Amer. J. Math. 107 (1985), no. 5, 1015-1033. MR 0805803 (87a:30055)
  • 4. Joseph Cima, and Alec Matheson, A nonlinear functional on the Dirichlet space, J. Math. Anal. Appl. 191 (1995), no. 2, 380-401. MR 1324020 (96g:46015)
  • 5. Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 259, Springer-Verlag, New York, 1983. MR 0708494 (85j:30034)
  • 6. Peter L. Duren, Theory of $H\sp{p}$ spaces, Pure and Applied Mathematics, 38, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • 7. Matts Essén, Sharp estimates of uniform harmonic majorants in the plane, Ark. Mat. 25 (1987), no. 1, 15-28. MR 0918377 (89b:30024)
  • 8. Jason James, and Alec Matheson, (in preparation).
  • 9. Donald E. Marshall, A new proof of a sharp inequality concerning the Dirichlet integral, Ark. Mat. 27 (1989), no. 1, 131-137. MR 1004727 (90h:30097)
  • 10. Alec Matheson, and Alexander R. Pruss, Properties of extremal functions for some nonlinear functionals on Dirichlet spaces, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2901-2930. MR 1357401 (96j:30003)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30H05, 30A10, 30D99, 49K99

Retrieve articles in all journals with MSC (2000): 30H05, 30A10, 30D99, 49K99

Additional Information

Valentin V. Andreev
Affiliation: Department of Mathematics, Lamar University, P. O. Box 10047, Beaumont, Texas 77710

Keywords: Dirichlet space, Chang-Marshall inequality, Baernstein star-function, extremal functions
Received by editor(s): August 1, 2003
Received by editor(s) in revised form: March 12, 2004
Published electronically: January 31, 2005
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society