334 CHAPTER 8 • REsIDUE THEORY
t Corollary 8. 2 Suppose that f and g a.re analytic functions defined in the simply
connected domain D , that C is a simple closed contour in D, and that f and g
have no zeros for z E C. If the strict inequality If (z) + g (z)I < If (z)I + lg (z)I
holds for all z EC, then Z1 = Z 9 •
Remar k 8. 5 Theorem 8.10 is usually stated with the requirement that f
and g satisfy the condition If (z) + g (z)I < lg (z)I, for z E C. The improved
theorem that we gave was discovered by Irving Glicksberg (see the American
Mathematical Monthly, 83 (1976), pp. 186-187). The weaker version is adequate
for most purposes, however, as the following examples illustrate. •
- EXAMPLE 8.26 Show that all four zeros of the polynomial g (z) = z^4 -
1z -! lie in the disk D2 (0) = {z: lzl < 2}
Solut ion Let f (z) = -z4. Then f (z) + g (z) = -7z- l, and at points on the
circle C2 (0) = {z: lzl = 2} we have the relationlf(z) + g(z)I ~ l-7zl + l-11=7(2) + 1<16 = lf(z)I.
Of course, if If (z) + g (z )I < If (z)I, then as we indicated in Remark 8.5 we
certainly have If (z) + g (z)I < If (z)l+lg (z)I, so that the conditions for applying
Corollary 8.2 are satisfied on the circle C2 (O). The function f has a zero of order
4 at the origin, so g must have four zeros inside D2 (O).- EXAMPLE 8. 27 Show that the polynomial g (z) = z^4 - 7z- 1 has one zero
in the disk D 1 (0).