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1 April 2005 Weighted Bergman spaces and the integral means spectrum of conformal mappings
Håkan Hedenmalm, Serguei Shimorin
Duke Math. J. 127(2): 341-393 (1 April 2005). DOI: 10.1215/S0012-7094-04-12725-3


The classical theory of conformal mappings involves best possible pointwise estimates of the derivative, thus supplying a measure of the extremal expansion/contraction possible for a conformal mapping. It is natural to consider also the integral means of |ϕ'|t along circles |z| = r, where ϕ is the conformal mapping in question and t is a real parameter (0 < r < 1 if ϕ is defined in the unit disk, while 1 < r < +∞ if ϕ is defined in the exterior disk). The extremal growth rate as r → 1 of the integral means which follows from the classical pointwise estimates is by far too fast. Better estimates were found by Clunie, Makarov, Pommerenke, Bertilsson, Shimorin, and others. Here we introduce a new method—based on area-type estimates—which discards as little as possible of the information supplied by the area methods. The result is a considerable improvement in the estimates of the integral means spectrum known up to this point.


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Håkan Hedenmalm. Serguei Shimorin. "Weighted Bergman spaces and the integral means spectrum of conformal mappings." Duke Math. J. 127 (2) 341 - 393, 1 April 2005.


Published: 1 April 2005
First available in Project Euclid: 23 March 2005

zbMATH: 1075.30005
MathSciNet: MR2130416
Digital Object Identifier: 10.1215/S0012-7094-04-12725-3

Primary: 30C40
Secondary: 30C85 , 32A25 , ‎32A36‎ , 46E22

Rights: Copyright © 2005 Duke University Press


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Vol.127 • No. 2 • 1 April 2005
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