22 Chapter^1Theorem 1.4 If n complex numbers zi, z 2 , ... , Zn (n ~ 2) are given
arbitrarily, then
(1.6-9)and the equality sign holds iff the ratio of any two nonzero terms is positive.
Proof The case n = 2 has been considered in Theorem 1.3, property 7.
Assuming that (1.6-9) is valid for n = k, we have
(1.6-10)Then for the case n = k + 1, we obtain, applying (1.6-5) and (1.6-10).
Jz1 + z2.+ · · · + Zk +zk+il = J(z1 + · · · + Zk) + Zk+1l
::=; lzi + · · · + zkJ + lzk+1l::=; Jz1I + · · · + lzkl + Jzk+1I
H;ence by mathematical induction, the property is valid for all n ~ 2.
If we assume that equality holds in (1.6-9), we get
Jz1J + Jz2J + · · · + Jznl = J(z1 + z2) + · · · + znl
::=; lzi + z2 I + Jza I + · · · + Jzn I
::=; Jz1J + Jz2J + · · · + lznlTherefore, equality must hold throughout, so that lz1 + z2J = lz1 I+ lz2J,
and if z2 ::/= O, it follows that the ratio zi/ z2 must be real and nonnegative,
i.e., zi/ z2 ~ 0, which implies that zi/ z 2 > 0 if z 1 of. 0. Of course, the same
reasoning may be applied to any other pair of nonzero terms.
Conversely, assuming, for instance, that z 1 of. 0 and Zk / z1 ~ 0 ( k =
2, ... , n), we have
This completes the proof.