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. Let 2f = {Dl9..., Dn) be a family of domains in the plane, containing the origin. We define a radial averaging transformation 01A on 2 by which we obtain a starlike domain D*. When 2 is such that the domains Du...9Dn are obtained from a fixed domain D by rotation or reflexion, StA becomes a radial symmetrization. One of the results we present is an inequality relating the conformai radius of D* to the conformal radii of Dl9..., Dn at the origin. This result includes, as particular, cases, the radial symmetrization results of Szegö [6] (for starlike domains), Marcus [4] (for general domains) and Aharonov and Kirwan [1]. The inequality for the conformai radii is obtained via an inequality fonconformal capacities. A number of applications in the theory of functions is described.

A radial averaging transformation, capacity and conformal radius