Weighted Bergman spaces and the integral means spectrum of conformal mappings

By H. Hedenmalm, S. Shimorin. 2004

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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 \phi'\(t) along circles \z\ = r, where phi is the conformal mapping in question and t is a real parameter (0 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 tip to this point.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/570bd059912b61c8df38853a03d7d83505b9fcfa

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Weighted integrability of polyharmonic functions (2012, background): “The notion of the critical integrability type is rather parallel to Makarov’s integral means spectrum in the context of bounded univalent functions [32] (see also [28]); there, however, a “sup” is used instead of an “inf” in the formula analogous to (2.”