Weighted Bergman spaces and the integral means spectrum of conformal mappings

Abstract

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.