The range of vector-valued analytic functions
Lemma. Let f : I ~ X be a path in a complex Banach space X satisfying f(O) = O. Let ~ >0 and 0< r < 1. Suppose that E is a closed subset o f the boundary o f A of linear measure 0 which does not contain a point z o, lz01=l. Denote D = A n K(r, co). There exists a continuous function ~ : A ~ X , analytic on A and having the following properties (a) f b (A)c f ( I ) + B , (X) (b) I[q~(z)l[ < 8 ( z E A D ) (c) ]If(l)-q~(Zo)[[ < e (d) ~(z) = 0 (z~E). Proof. By the Mergelyan theorem for vector-valued functions [1] there exists a polynomial P: C-*-X such that IF f ( z ) -P(z)][ <e/4 (zCI). Putting Q ( z ) = P ( z ) P ( O ) it is easy to see that there exists a neighbourhood U of I such that Q ( U ) c f ( I ) + +B, (X) . Let Vc U be an open neighbourhood of I -{1} , containing the point 1 in its boundary and bounded by a Jordan curve contained in U. By the Riemann mapping theorem there exists a one-to-one analytic map q~ from A onto V which [4, p. 282] has an extension ~ which is a homeomorphism from A onto V. Composing