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.