Weighted Bergman spaces and the integral means spectrum of conformal mappings

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 < r < 1 if phi is defined in the unit disk, while 1 < r < +infinity if phi 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 tip to this point.