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.
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