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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Dirichlet integral and star-function inequalities


Author: J. R. Quine
Journal: Proc. Amer. Math. Soc. 96 (1986), 249-254
MSC: Primary 31A05; Secondary 30C55, 30D35
MathSciNet review: 818454
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Abstract: Let $ \lambda (z)$ be a smooth function in an annulus, $ \tilde \lambda (z)$ its symmetric rearrangement and $ {\lambda ^ * }(z)$ its star-function. A formula is proved relating $ \Delta {\lambda ^ * },\Delta \lambda $, and the Dirichlet integrands of $ \lambda $ and $ \tilde \lambda $. The formula shows the relationship between Dirichlet integral inequalities and the subharmonicity of $ {\lambda ^ * }$ for subharmonic $ \lambda $, and gives an explicit formula for $ \Delta {\lambda ^ * }$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0818454-8
PII: S 0002-9939(1986)0818454-8
Article copyright: © Copyright 1986 American Mathematical Society