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32 S. W. DRURY
and if h1 is not an endpoint of [−1, 1],
0= ∂g =At(|z|2h1 −Rez)−Bth1
∂h1
with A and B essentially as before. We find that t(Rez)(h1 −h2)2 = 0 by eliminating A and B. This forces h1 = h2 and again the situation is essentially one-dimensional, so we can assume that h1 is an endpoint. Similarly we can assume that h2 is an endpoint. But then h21 = h2 = 1 and
g(t,h1,h2)= (t|1−zh1|2 +(1−t)|1−zh2|2 −r
and the minimum is clearly attained either for t = 0 or for t = 1. The proof is
complete.
References
[1] H. Wielandt, On the eigenvalues of sums of normal matrices, Pacific J. Math. 5 (1955), 633–638.
(S. W. Drury) Department of Mathematics and Statistics, McGill University, Burnside Hall, 805 ouest, rue Sherbrooke Street West, Montre´al, Que´bec, Canada H3A 0B9
E-mail address: drury@math.mcgill.ca
