Some Uniqueness Results in the Theory of Symmetrization

Abstract

1. The symmetrization methods developed by P61ya and Szego have proved to be of fundamental importance in several branches of Mathematics and Mathematical Physics. In the Theory of Functions the most important results appear to be those such as the fact that the module of a doubly-connected domain is not decreased by circular symmetrization. The proof of this depends on considerations involving the Dirichlet integral. As P6lya and Szeg6 have remarked, the method of proof used by them offers little possibility of deciding when the module indicated above is actually increased and of solving similar problems. For the Function Theoretic applications it is important to know the answer to these questions in order to obtain certain uniqueness results. The object of the present paper is to prove some results of this type. 2. Let D be a doubly-connected domain which lies in the w-plane and which is bounded. It will have two complementary continua C1 and C2 which we assume to be non-degenerate and of which just one, say the former, will be bounded. By circular symmetrization we obtain from D a second doublyconnected domain D* as follows. Let the intersections of I w i = R with C1 and C2 have respectively angular Lebesgue measures 11(R) and 12(R). Let C1* be the set -121(R) M(D). The question which we will first settle is that as to when the equality sign can occur here. If equality does occur D* is the image of the circular ring ri < z I < r2 by a regular univalent function w = f*(z). We again suppose that I z = ri corresponds to the boundary component of D* in the continuum C1* . We are now ready to state our first principal result. THEOREM 1. The equality M(D) = M(D*) occurs if and only if f*(z) e af(e tz) with a, j3 real.