1550251515-Classical_Complex_Analysis__Gonzalez_

586 Chapter^8

lf'(e)I:::; 1 -lle12

Equality occurs iff F(z') = az', or f(z) = a[(z -e)/(1-cz)] with !al= 1.

The following theorem improves the first estimate in Schwarz's lemma

when the origin is a zero of multiplicity k > 1.

Theorem 8.38 Suppose that:

1. f is analytic in lzl < 1

2. lf(z)I :::; 1 for lzl < 1

3. f(O) = f'(O) = · · · = J(k-l)(O) = O, j(k)(O) -/= 0

Then we have the estimates

for lzl < 1 and (8.13-8)

Equality holds iff f(z) = azk where !al = 1.

Proof Because of hypothesis 3 the Cauchy-Taylor expansion for f(z) about
the origin has the form

J(k)(O) k J(k+l)(O) k+i

f(z)=-k-!-z+(k+l)!z +···

If we define

j(k)(O) j(k+ll(o)

g(z)= _k_!_ + (k+l)! z+···

so that g(z) = f(z)/zk for 0 < lzl < 1 and g(O) = J(k)(O)/k!, the remainder

of the proof can be carried out as for the lemma.
As an application of Corollary 8.18 we shall prove the following property,
which is useful in the theory ~:f entire functions (Selected Topics, Chapter 3).

Theorem 8.39 (Borel-Caratheodory Theorem). Suppose that f is

analytic in a region G containing the closed disk lzl :::; R, and let

M(r) =max IJ(z)I,

lzl=r

Then for 0 < r < R we have

A(r) = maxRef(z)

lzl=r

M(r):::; R

2

~rA(R)+ ~=~lf(O)I (8.13-9)

Proof If f(z) = e = c 1 + ic 2 (a constant) in G, then (8.13-9) becomes

2r R+r

lei :::; R -r ei + R - r lei

or -e1 5 lei, a trivial inequality.