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

 

Area extremal problems for non-vanishing functions


Author: T. Sheil-Small
Journal: Proc. Amer. Math. Soc. 139 (2011), 3231-3245
MSC (2010): Primary 32A36; Secondary 30J99
Published electronically: April 6, 2011
MathSciNet review: 2811279
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Abstract: We consider the problem of finding the extremal function $ f $which minimises the Bergman space $ A^2 $norm for the class of non-vanishing functions whose first $ n+1$ Taylor coefficients are given.$ $We define an analytic function $ K$ in terms of $ f$ and show that the functions $ K$ and $ f$ satisfy a certain differential equation. This equation yields a set of relationships between the area moments and the circle moments of $ \vert f\vert^2$, which in particular shows that the outer part of $ f $is a polynomial of degree at most $ n$.


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Additional Information

T. Sheil-Small
Affiliation: Department of Mathematics, University of York, Heslington, York, YO 10 5DD, United Kingdom
Address at time of publication: P. O. Box 60681, CY8106, Paphos, Cyprus

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11031-1
PII: S 0002-9939(2011)11031-1
Received by editor(s): March 14, 2010
Received by editor(s) in revised form: July 16, 2010
Published electronically: April 6, 2011
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.