128 9. SECOND SERIES OF IMPLICATIONS
and by (9.4.4)
l(FT F - I)zl^2 = l(IJ(l)l^2 - 1) x + Re(f(l)f(i)) Yl^2
+ IRe(f(l)f(i)) xl^2 + l(lf(i)l^2 - 1) Yl^2 ~ 9(L^2 -1)^2 lz l^2 ,whence
(9.4.5) l(FT F - I) z l ~ 3(L^2 - l)lzl < 3(L~ - l)lzl = lzl
by (9.4.3). Thus the eigenvalues of pT F - I lie in (-1, 1) and the eigenvalues of
p T Fare positive. In particular, F(B) is an ellipse with semiaxes )q, >-2, where
(9.4.6) Jl - 3(£2 - 1) ~ >-1 ~ >-2 ~ Jl + 3(L^2 - 1),
and there exist angles a, j3 such that
F(eio:) = >-1 eif3, F(iei°') = i.-\2eif3.
More specifically, F(z ) = T(eiez) where e = j3 - a and T : R^2 -+R^2 is the linear
transformation with
T(eif3) = >-1 eif3, T(i eif3) = iA2 eif3.
Here
(9.4.7) IT(z) - zl = l(T-I)zl ~ blzl
where
(9.4.8) b = 1 - Jl - 3(£2 - 1) ~ 3(L^2 - 1).
Finally, since
f(l) = F(l) = T(eie), f(i) = F(i) = T(i eie),
(9.4. 7) implies that
(9.4.9) lf(l) - eiel ~ b,Then for lzl ~ 1 we obtain
lf(z ) - ei(} z l^2 = lf(z) e-ie - z l^2
= IRe(f(z)e-ie - z)l2 + IIm(f(z)e-ie - z)l2
= IRe(f(z)e-ie - z)l2 + IRe(i f(z)e-ie - i z)l2.Next the fact that IJ(z) I ~ L , (9.4.9), (9.4.4), and (9.4.8) imply that
IRe(f(z)(e-ie - z))I ~ lf(z) l le-ie - f(l)I + IRe(f(z)f(l) - z)I
~Lb+ (L^2 - 1)(1 + lzl^2 + lz - 112 )/2
~ 3(L + l)(L^2 - 1)and similarly
IRe(i f(z)e-ie - i z)I ~ IJ(z)l Ii e-ie + f(i)I + IRe(f(z)f(i) + iz)I
~Lb+ (L^2 - 1)(1 + lzl^2 + lz - i l^2 ) /2
~ 3(L + l)(L^2 - 1).We conclude that
lf(z) - eie z l ~ 3J2(L + 1)^2 (L - 1) < a(L - 1),
where a < 20 from (9.4.3). D