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Proceedings of the American Mathematical Society
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
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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.


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Additional Information

Valentin V. Andreev
Affiliation: Department of Mathematics, Lamar University, P. O. Box 10047, Beaumont, Texas 77710
Email: andreev@math.lamar.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07712-9
PII: S 0002-9939(05)07712-9
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.