By Lawrence Zalcman

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Thus S2 \ K . can be Now for -36- I! f21~ = IIf-f 11~ _< II fl~ + II flllr Ilfl~ + IIf-f211r = __< llf lk + Ill lit + llf2 lit _< 2 +~ 1 . 7. f E C~) Let ~ K be analytic on c Proof. r . This is the content of be a compact s u b s e t of A \ K l~r f(~)d¢l where is a universal and satisfy . Let IIfll~_< 1 K is bounded by a finite number G(K) = S2 \ K . as before, is smooth on all of we can assume that E CR~ ) Then constant. We may assume that O(z) . _< cy(K), of smooth closed curves and that Let it is replaced by an absolute constant, to extend to K actually, be such that f G(z) = 1 for Moreover, A .

Be a compact set in the plane. K , f(~) = 0 , and If'(~)l _< 2~¥(K) f(z) = ~ A n z -n Let f Re f(z) < a , where . at Then the admissible ~ . n=l functior~ fl(z) = f(z)[f(z) fl(z) = (-Al/2~)z-1 + ... 2. Let - 2a] -I at a I! f ll,,. 1 If' (a)ip(p-~,(K)) Proof. 10, we may assume that Then I = to K . a 6 G(K) , and let the distance from . If' Thus as required. be compact, K - . has the expansion Let f ~ be analytic off _< 2y(K) K be K , . has a boundary con- sistiug of finitely many smooth closed curves.

K , such that for each we m a y a s s u m e that is obvious, n and 3K n nK n = K . 10. n nor on K It is enough to prove that [f'(~)[ < c[y(Kn) + ¥(Kn*)] for f admissible N o w let modify S2 f without n for K U K* . be fixed and so that it becomes disturbing f as ~bove be given. We can a smooth f u n c t i o n on all of its values on $2' \ (K n U Kn*) . -31- Let us assume this modification accomplished. Let f(z) + f(I/~) , y_(z) = f(z) - f(i/~) ~+ . Then and are smooth functions analytic off K n U Kn* Also, i s imagirmz-E there.