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.
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