A General Class of Problems in Conformal Mapping.

belongs to class S if f(z) is regular and schlicht in zI < 1 and a, = 1. Let R be a closed set lying in zj < 1, and let *,1,(r) be a measure function defined in the space R. Given an integer n there is a number M M. such that If()-(z) < M for all z C R and for v = O, 1, 2, ..., n when f(z) belongs to class S. Let F,(ro Ro, ..., rny ) denote a complex-valued function which is continuous together with its first-order partial derivatives in an open set containing the closed set ,, _ Mt (v .= O, 1, ..., n). -Given the function's F,, F2, ... Fm and the mneasure functions '1, I2, ..., Wm, let