Transformations of domains in the plane and applications in the theory of functions

Moshe Marcus
1964 Pacific Journal of Mathematics  
In this paper we shall consider a family of transformations S n (n = 1, 2, •) operating on open or closed sets in the complex plane s S n is defined relatively to a fixed point called the center of transformation, and it transforms an open set into a starlike domain which, for n > 1, is also %-fold symmetric with respect to this point. Therefore, for n > 1, S n may be classified as a method of symmetrization. This method of symmetrization was already defined by Szego [4] for domains which are
more » ... arlike with respect to the center of transformation. The definition of S n will be extended (in the way usually used for symmetrizations) so that S n will operate also on a certain class of functions and a family of condensers, in the plane. It will be proved that S n diminishes the capacity of a condenser and this result will be used in order to obtain certain theorems in the theory of functions. his guidance and help in the preparation of this work. 613 614 MOSHE MARCUS Now, the set obtained from Ω by the transformation S n = S n (z 0 ), with center z 0 is defined as follows: ( 4) S n Ω = {z | z -z 0 = re iφ y 0 ^ r < R {n) (φ), 0 ^ φ < 2π} . If instead of Ω we have a compact set H, which has an interior point z 0 , we define: It is easily verified that S n Ω is a simply-connected domain and that S n H is a connected compact set. Both sets are starlike with respect to z 0 . We shall extend the definition of S n over a family of functions & which will now be defined. A non-constant real function g(z) belongs to ^ if it is continuous over the extended plane z, if it takes its maximum value at infinity and if its minimum is assumed on a set of points, the interior of which is not empty. Let g(z) be a function of *& and let m and M be its minimum and maximum values, respectively. We define the following sets: for m < c ^ M .
doi:10.2140/pjm.1964.14.613 fatcat:qrh7g2sbtzcrzeqwcfc32yteem