On the distortion in conformal mapping of variable domains

Formanyproblems inconformal mapping it is of importance to have precise quantitative information concerning the variation of the mapping function of a region onto a circle, under a deformation of the region. A number of estimates for this variation in terms of parameters, which measure the deformation and the smoothness of the boundary of the region, have been given in the literature. A survey of such results is contained in [13] and [i1 ](2). While the degree of the change in the mapping function itself has been investigated for various general configurations, the corresponding question concerning the distortion, i.e. the derivative of the mapping function, has been studied primarily for nearly circular regions. The object of the present paper is to obtain such estimates of the distortion for arbitrary regions under quite general hypotheses. The paper is divided into two parts. The question considered in the first part arose in connection with the following theorem [8, p. 364]. Suppose C and Cn (n= 1, 2, * * * ) denote closed Jordan curves which are represented by the equations