10

STEPHEN SEMMES

h : A —• Bn is defined by h = p~l o y, then ft is holomorphic away from the

origin, because dp is a complex-linear map of S1 to T1. Because h is bounded it

is also holomorphic at the origin.

Let's determine h'(Q). Because g'(0) = a/'(0), we have that g(X) — f(aX) =

0(|A|2)

as A —• 0, and so g(X) - p(Xav) =

0(|A|2)

by definition of / . Hence

h(X) — Xav =

0(|A|2)

also, and so h'(Q) = av.

Now we are essentially finished. Define h : A — * C n by h(X) = A-1/i(A)

when A ^ 0, h(0) = hf(0). Then h is holomorphic and lim \h(X)\ 1, since

MA) C B

n

, and so h(A) C Bn. Because h(0) — av E dBn we conclude that /i is

constant. Hence h(X) = Xav for all A, and so

g(X) = p(h(X)) = p(Xav) = f(aX),

as desired. This completes the proof of Theorem 2.5.