Compressibility and Uniform Convergence.

.z I < p(e), E < p(E) < Po, and p(c) depends not only on e but also on w(z). The function p(E) clearly decreases monotonically with e; it also approaches zero with e, for the transformation z(w), although multiple-valued, is continuous. If now E(e) lies in Jz < E, E < Eo, then the convex hull of w(E(e)) contains the point w+, and the m antecedents z+ of w+ in jz < Po defined by equation (5) lie in Iz < p(E). Theorem 6 is established. This convex hull contains also every w-, so every antecedent of win Iz < Po lies likewise in jz I < p(E). Theorem 5 is the special case m = 1; for this case, we may take p(e) = E if e is sufficiently small, namely, if W(E) is convex, but it may be appropriate to distinguish p(e) and e for larger values of E, namely if w(z) is univalent in Iz I < E while W(E) is not convex. A sharper result than Theorem 6, for the case that the E(e) are mutually similar, with 0 as center of similarity, is given in reference 1, end of section 1. * Research supported (in part) by the Office of Naval Research and by the Office of Scientific Research, Air Research and Development Command. 1 Motzkin, T. S., and Walsh, J. L., "Zeros of the error function for Tchebycheff approximation" (to appear). 2 Walsh, J. L., "On the shape of level curves of Green's function," Amer. Math. Monthly, 44, 202-213 (1937).