An extremal function for the Chang-Marshall inequality over the Beurling functions
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- by Valentin V. Andreev PDF
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Abstract:
S.-Y. A. Chang and D. E. Marshall showed that the functional $\Lambda (f) =(1/2\pi ) \int _0^{2\pi }\exp \{ |f(e^{i\theta })|^2\}d\theta$ 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
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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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 2069-2076
- MSC (2000): Primary 30H05; Secondary 30A10, 30D99, 49K99
- DOI: https://doi.org/10.1090/S0002-9939-05-07712-9
- MathSciNet review: 2137873