1550251515-Classical_Complex_Analysis__Gonzalez_

588 Chapter^8

Exercises 8.5

1. If f is analytic in lzl < 1 and such that IJ(z)I < 1, show that (G. Pick

[28]):

(a) I f(z)-f(w) I::=; I z-~ I for lzl < 1, lwl < 1

1-f(w)f(z) 1-wz

lf1(z)I 1

(b) i - lf(z)l^2 ::=; l - lzl^2

Equality holds if f(z) = (az + b)/(bz +a) with aa - bb = 1.

The above is the invariant form of Schwarz's lemma, also known as

the Schwarz-Pick theorem. It can be stated by saying that an analytic

mapping of the unit disk into itself decreases the hyperbolic distance

between two points as well as the hyperbolic length of an arc.

(c) Assuming that J(1/ 4 ) = 0, use part (a) to estimate 1!(2/a)I, and use

part (b) to estimate If' (2/a) I·

2. If f is analytic and lf(z)I < 1 for lzl < 1, show that

lf(O) + lzl

lf(z)I:::; 1 + IJ(O)llzl

3. If f is analytic and IJ(z)I < 1 for lzl < 1, and if f(O) = c, show that

I ( ) I

< I I 1 - lcl2

f z -c - z 1-lcllzl

for lzl < 1.

4. Suppose that f is analytic and Ref(z) > 0 for lzl < 1, and that f(O) = 1.

Prove that

1 - lzl < lf(z)I < 1 + lzl

1 + lzl - - 1 - lzl

for lzl < 1. Hint: Let F(z) = [f(z) -1]/[f(z) + 1].

5. Suppose that f is analytic and IRef(z)I < 1 for lzl < 1, and that

f(O) = 0. Show that

4

I Ref(z)I ::=; - Arctan lzl

7r

and

I Imf(z)I ::=; ; ln ~ = :::

for JzJ < 1.