Math 220C - Lecture to- -- -

May 1,2020

Last time- --

Little Picard f : a → a entire & omits o,n =, f- constant

.- -

--

Step . I = I Cn + cos it cost g) , g entire

• A = { m ± ÷ log Cn + Tnf) : M E Z in C- 2>o) .

• A gives rectangles Rmn whose sides are I 1.

• g omits the values in A

^

gtm in""

jtmtlint'

- - - - -ora¥=¥÷ks"+r¥sz .

Arm , I l

l ll l#

m mth

tdemark : If I is any disc of radius 2 , In A FOI . V vertex of Rmn1

Indeed if a center of D ,lets c- Rmn .

=3 d la,V) (T.cz

.

I> V E A Cd. 2) N A

,. ⇒ I n A f- OI .

Conclusion (m g cannot containany disc ↳ of radius 2

.- - -

:

Iepts (Bloch). Let 1 be the unit disc . Define

G CI) = { f : holomorphic in some G Z E}.

Theorem If f- E G CE).

, f' cos =L .⇒

- -

Imf contains a disc of radius 13 .

For a constant 13 fo .owhioh does not depend off

In fact,we can take 13 = Ira - 2

.

Finish the proof of Picard Assume g # constant =' g'(Zo) -40 for some Zo

.- - - - -

whos 2 . = o . otherwise work g. ( 2- 1- Zo) . Thusg'co) # o

.

Let f = 9 ( RZ) Bloch-

. => f ' co) = 2.Is Im f 2 disc of radius 13 .

> o .R g'co)

thin=> Img contains a disc of radius 13 . R . I g

'fo>I

. > 2 . For some R.

This is a contradiction with the Conclusion on the previous page .

Remark II In Bloch's thin , the disc may not be at o.or fees

.

- - -

HI there are versions of the thin where the disc.is

the biholom orphic image of a subdomain m I.

this is done in Conway .

13 = Izz ¥.613 in the book

.

We will give 13 = Irs - a =.12

. The exact optimal 13 isunknown

.

Remand Let F = f f E G CES , f- ' cos - I}.

• Lf = { the largest radius of a disc contained in fcs, ]

• Bf = { the largest radius of a bi holomorphic disc in fca)}.

• L = Inf Lf I Landau

• B = snf Bf£ Bloch

.

• 433 S B L.472

•. 5 L L L

. 544

• conjecturally B=r- . r(})⇒=.

47 ,

real• We show L Z p = f- Fs - 2 = . 121

Proof of Bloch 's theorem f- ' ios =p,f -E GCE) =3 Imf 2 Disc radius 13 .

-- - -

Question How do we find a disc contained rn Im f ?- -

LemmeA- Assume f E G ( E). .C, bounded ,

a EG G-

- -

i-• Z

Tat p =

zm.gg ! fin- feast ⇒ Imf contains

the disc A- ( fca), p) .

✓

troof Let H = f CG) .

Let R = d ( flag,att)

.

> o) t '

mafia,

which makes sense since 2h compact .

- 00

( it bounded.

H E f CE) bounded) .

⇒ ⇐ , , ,a,, ,, , ,mf . yya.fm. ,→n

.

TO ,

we only need R If . → Imf 2 I ( flag,r) .

W = fit)

Indeed,let us C- 2h

, R =D cfca) ,w).

If we prove w = f- CZ) , 2- C- 25

We are done since

R = d ( Fca) , Fez)) Ip .by the definition of p .

The fact that Z C- 26 is the open mapping theorem . The details :

Since U E 2h ⇒ F FCZN) → W,Zn E G . Not Zn → z m G-

We show 2- E 26.

Then W = f- CZ) . d 2- C- 26.

,n> so

.

["" " "H''ng thin .

If 2- Haa then 2- EG .

⇒ F small disc I E G t z

centered at Z which contains Zn's . then f- ( Es is open in H = f- CG).

and contains '

w = f- CZ) .and also Fczn) for n large . But

W E 2h so this is impossible .

Question :Can we apply LemmaA to prove Bloch's theorem ?

.

- --

Steps : I first under assumption C*)- -

remove assumption (*) .

In Tel we get larger disc with center at frog.

-o

(n ④ ,we lose control of the center

, decrease the radius by # ,

but no assumption is made.

A-ssu-mph.org (*) sup if'

Czsl I 21 f-'lost

.

⇐ 17k£> Is 2 if floss ,.

,

"

12-111

Slept :theorem 2e't f E G CI)

.

not constant,

- -

with property ( *) .

The n.

image of f contains the disc A-R( f co) ) where R = (3 - 2K) l F' cost .

Pzoeef The goal is to estimate lfcz) - f- cost for all 12-1=12.-& apply

Lemma A,

W LOG f- Co) = o

Let F Cz) = f-Cz) - Z f- ' co). By FTC ,

we have

FTC

[ z

la) I 2- f' coil - ifczsl E IF 't't =/ ) F' Cws - *' co, dwy ! I It! !I o

triangle = / fo"

f-'

Cz t) - F'cos 2- dtfinequalityE f

.

"

If'

lzt) - f- ' cost . 12-1 dt .

By Cauchy 's formula , we obtain

(2) ( F' sets - F' cost =/ I t' - FYI us!131=1

.

± ÷ . I ,t; #t d 's.

3. ( S - zt)131=7 = -f by assumption (*)

E ÷ I izitds

131=1 7 - 121T

=# . 21 F' lost. !Z#,I . 2€

Thus by G) : I 2- f' coil - lfczsl € f.

"

If'

lzt) - f- ' cost . 12-1 dt .

(2) I

f fo 2 I f- ' lost . Htt . lzldtI - 121T

E 2/17' cost 12-1 ? If . fo't dt-

z Yy= tf ' cost . R-

for 12=12-1 -

I -R

This gives 1712-71 2 If' cost - ( R .

- ,R÷p ) . For 12-1=12 .

The function R - RI is optimized at R*= n - I Its value isI - Rz

"

3 - 252.

Thus for 12-1 = R* ,

[emma AI f- CZ) I 2 ( 3 - 252) 174031.

=> theimage of f

contains F ( I (o ,R*)) 2 disc I ( o,13 - 2K)) f-

'

cool) as needed.

DV '

Remark the fact that we work in D Cgi) is not smsortant . Indeed ,- -

⇐lay tf F is holomorphic in I co,R) and .

Supt 742.71 E 2 If' cost,

12-1=52.

then Im f contains I (f- co) , 13 - 2K) R tf ' cost) ."D

thoof Work with f ( z) = f- CRZ) . in the disc II lo, D .

& apply the theorem .

I

SIP We seek to remove the assumption

I f-'G-31 I 2 If

'

cost t 12-1=1. We do know that

if this holds we obtain a disc of radius ( 3 - 252) 174031.

Thus

larger f- ' co) - larger disc .\

If we wish to fend a large disc we should optimize lflcos , while

keening the Image of f fixed.

Key Idea Work with foot where E- C- Aut CE).

- -

We keep Image the same but modify the derivative.

.

Remark In other words,we will derive Bloch from ID

- -

applied to a suitable function foot .

Proof of Bloch Let f e GCI ) .

We define-- --

A- = { IT = Eta o Rot :) where Ea = ⇐r

is an

r - IZ-

the

automorphism of I and Rot denotes rotation . Then A isaf

group of automorphisms of JL .

Let F =/ foot . OI c- A} .

'

.

If I = fo QT and

I = OI, o'

Roth then

CE lo) =L, IET

'

cost = n - 1212

( *) It ' cos I = If'

cohost . of'

cost = If ' cast . ( e - lat).

Let an = sup IF' cast . ( r - laid)

. (t) .

I al II

Claim A- t F E F ⇒ IF ' coil E M. by (*) and (f) .

Remark As I c- I => I f- ' cost s ha- -

-

Cz B t F E F => IF'

Cz) l E M- For 12-1<1.-

n - s-212"

PII For all 13 ,F o oIp E F for all FEI since it is

a

group -

claim A ⇒ IE o top)'

loll EM.

By direct computation , 1€ - Ep)"

cost = IF 'C&ploDl . top' cost ='

= It'

Cpsl . µ - Ipf)Therefore IF 's post - Cr - ' psid ) Im ⇒ Elam's

.

Ciempleteng the pzof#310£ Let a be the point where the maximum

(t) is achieved. Thus

,

IM = f- ' cast - Cn - lat)

Let F = f - OIL . By (*) , we have IF' Coy = ha

.

By claim B,since E E F

,

we have

taIE ' Cz) I I - I 2M = 2 I F' lost if I 2- I =L1- 12-12 Fs

By Part Id, applied to I

,Im F CD) = Im f- Cb) contains

(m F ( I ( o , Jg)) Z disc of radius (3-252) ÷ IF to> I.

1Part ④ = ( 321 - 2) in 2corollary

Remark ←I (3¥ - 2) If 'cost .

following claim As

this completes the proof of Bloch & Little Picard along with it .