1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 583

then g(z) is analytic in izl < 1, and

g(z) = f(z)

z

for z '=f 0 and g(O) = f'(O) (8.13-2)

Let z be such that izl < 1, and chooser such that lzl < r < 1. By the

maximum modulus theorem, lg(()I attains its maximum on ICI ~rat some

point (or points) on its boundary ICI = r (Fig. 8.7). Hence if we take the

hypothesis (2) into account, we have

lg(z)I ~max lg(()I =max lfl~~)I ~ ~ (8.13-3)

The left-hand side of (8.13-3) is independent of r and the inequality holds
no matter how close r is taken to 1. Hence we obtain

lg(z)I ~ 1

for any such that izl < 1. In view of (8.13-2), it follows that

lf(z)I ~ lzl for^0 < izl <^1 and lf'(O)i ~ 1 (8.13-4)

However, the first inequality in (8.13-4) also holds for z = 0, since f(O) = 0
by assumption.

If lg(z)i = 1 at some point zo of the open region izl < 1, then lg(z)I

attains its maximum value at z 0 • By Theorem 8.29 this is impossible
unless g(z) is a constant; i.e., g(z) = a, where a is a constant such that
iai = 1. In this case we have

f(z) = az (8.13-5)

On the other hand, if (8.13-5) holds with iai = 1, we have lf(z)I = lzl for

all z and f'(z) =a, so that IJ'(z)I = 1, in particular, lf'(O)I = 1.

y

x

Fig. 8.7