The Coefficients of Schlicht Functions

Abstract

(0 a < 1) (4.8)'-r(a, b) = S (Ua, ub; p, q, r) (O : b < 1). (4.8)" The remaining values q(1, 1), r(a, 1) and r(l, b) are determined by the values given in (4.8), taking account of the continuity of q(1, ) and r(l, ). Thus the model set (p, q, r) is uniquely determined by S. This completes the proof of the theorem.In terms of (p, q, r) the corresponding model set (p, 4, i) may be given as follows for (a, b) e E' X E' using (4.8).p(1, 1) = p(1, 1) (4.9)' q(1, a) a) + Ja (t-a)dp(1, t) = q(1, a), -fap(1, t)dt (4.9)' r(a, b) = r(a, b) -J;aq(s, b) ds Jbq(s, a) ds + f bp(St) ds dt (4:9) "' COROLLARY 4.1. A necessary and sufficient condition that S(u, v; p, q, r) -Ofor every u, v in IL2 is that the right members of (4.9) vanish for every (a, b) in E' X E'.

The Coefficient Problem For Multiply-Connected Domains

“…We define the degree of P, deg P, to be unity, and we define 2, if 9 belongs to an interior point of 9, and we see that deg 0 is the same whether a is considered as belonging to 9) or to W.…”

“…If v is a uniformizer at q, (q) = 0, thenis a uniformizer at q, and we have 9 log + regular terms (near q); g = log i + regular terms (near q);…”