April 8 , 2019

Maximum modulus principle ( finalversion :) Let G be a region in Cl

and f an analytic function on 6.

suppose that F M t finalSAME M

Fae do 6 .

Then I fall E M t Ze G.

-

EIA SAI =

f.ingots up {SHI: teh Bla

, D)

at G fi 6 -7 IR

On a =D.

16 = {DG

, if G bounded

86 Us -3 if6 unbounded

proof: Let so ,let°

H = { te G : ISHII > Mtc }Since Iflis continuous, H is open(

must show that His emptySince finalSAY E M fat dog

,

there is Blair ) t IfAll s Mtf

At EG A Bca , p ).

It follows that Face.

The same is true if G is

unbounded I a = a.

It follows that It is bounded

and I is compact.

This means that the second version

ofthe maximum modulus principle

applies.

Observe thatfor ze OH,

I fall = Mtf since I C

{ z i 18GHz ME }It follows that It -

- 01,

on

f is constant. But if f isconstant, He $ by assumption,

This completes the proof.

It is time to start reapingthe

rewards of maximum principleand there are

many!

Schwarz 'slemma :

Let D= { ti Hk I 3 and suppose

that f is analytic in D w/

i) IfAtl El ,

TED

ii ) f 61=0Then If

'

lol HI a HAVE HItf Ef D

Moreover , if If'

lol f- I on

if ISAY = 124 forforec- D

,

flu ) = e w tf u C- D.

To provethis

, define

g:D -71C by gGI=f¥,

*gcokjaf.toTheng

is analytic in D. By the

maximum modulus principle,

Ight I E f-'

if HEP,

OL PL I.

Letting in → I ⇒

Ig Call E I ⇒ I fall EH

I 184011 = Ig loll E I.

If If All= HI for some x-D

,

ZEO on

if 01ft ,then

g assumes its max inside D

⇒ gate⇒ feel = c E & we are

done !

we will now usethis to classify

all conformal maps of the diskto itself.For law

,define

each - Eff ,

analyticfor KK ki !

half . act ) ) = t = 4- a ( la HI )

calm ion

So la : D → D A

Hale in the'Ia÷.the:÷. f- I .so

Ya CdD) = OD.

In summary, if tall I :

i ) da is a I - I mapping of D

ii ) he is the inverse of da

iii ) la : 8D → 8D

idealat = O u) da Cott - lap

vile ! cat -

- G - lait)quotient rule

Suppose that f is analytic on D

w/ KAMEI .

Assume that laid

& flat =L.

Let g-

-

Logofat :D → D

geol = I (flat ) =Is(a) = o.

By Schwarz 's lemma

Ig'Coll El

.

We can go farther.

g of @of)'

Checo

's) ' field

= @of Jca ) CI - tail

= b'Kyla ) Chai )= I - lat

'

⇒flat

⇒ Is I EI - la 12

w/ equalitywhen lg4olH

Schwarz I⇒

JAI -

- ha Ceda GI ),

left

for Kk I.

Theorem : Let f :D -7 D be a It

analytic map of Doa to D &

flat -

-

O,

Then F c w/ left 7

f- cha .

proof: Since f is H,I g

t

g I fHh=z ⇒

g'

Col f'la ) = I.

Byabove

,

IS 'M ECI - latteIg

'loll El - lal

'

Cgeol - a )

⇒ Is'

Call = Ci - lait'

⇒ f = ela,

let =L.