Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells...

Majorization-subordination theorems for locally univalent functions. IV

A Verification of Campbell’s Conjecture

Roger W. Barnard, Kent PearceTexas Tech University

Presentation: May 2008

Notation

{ : | | 1}z z D

Notation

( )DA

{ : | | 1}z z D

Notation

Schwarz Function

( )DA

: , | ( ) | | |z z on D D D

{ : | | 1}z z D

( ) DA

Notation

Schwarz Function

Majorization:

( )DA

| ( ) | | ( ) | | |f z F z on z r | |f F on z r

: , | ( ) | | |z z on D D D

{ : | | 1}z z D

( ) DA

Notation

Schwarz Function

Majorization:

Subordination:

( )DA

| ( ) | | ( ) | | |f z F z on z r f F

| |f F on z r

f F for some Schwarz

: , | ( ) | | |z z on D D D

{ : | | 1}z z D

( ) DA

Notation

: Univalent Functions : Convex Univalent Functions

Notation


: Linearly Invariant Functions of order U

1 2, K= = U S U

Notation


: Linearly Invariant Functions of order

Footnote: , and are normalized by

U

1 2, K= = U S U

U 22( )f z z a z

Majorization-Subordination Classical Problems (Biernacki, Goluzin, Tao Shah,

Lewandowski, MacGregor)

Let F S



Let

A. | |If f F on find r so that f F on z r D

F S

(1967) : 2 3M r



Let

A.

B.

| |If f F on find r so that f F on z r D

F S

| |If f F on find r so that f F on z r D

(1967) : 2 3M r

(1958) : 3 8TS r

Majorization-Subordination Campbell (1971, 1973, 1974)

Let F U


Let

A.1

1

, | | ( )

( 1) 1( ) 1( 1) 1

If f F on then f F on z n

where n for

D

F U


Let

A.

B.

1

1

, | | ( )

( 1) 1( ) 1( 1) 1

If f F on then f F on z n

where n for

D

F U

2

, | | ( )

( ) 1 2 1.65

If f F on then f F on z m

where m for

D

Campbell’s Conjecture Let

2

, | | ( )

( ) 1 2 1 1.65


where m for

D

F U

Campbell’s Conjecture Let

Footnote: Barnard, Kellogg (1984) verified Campbell’s for

2

, | | ( )

( ) 1 2 1 1.65


where m for

D

F U

1K= = U

Summary of Campbell’s Proof Let and suppose that so that for some Schwarz

F U f F f F

Summary of Campbell’s Proof Let and suppose that so that for some Schwarz Suppose that f has been rotated so that satisfies

F U f F f F

(0)a f 0 1a

Summary of Campbell’s Proof Let and suppose that so that for some Schwarz Suppose that f has been rotated so that satisfies Note we can write where is a Schwarz function

F U f F f F

(0)a f 0 1a ( )( )

1 ( )a zz z

a z

Summary of Campbell’s Proof Let and suppose that so that for some Schwarz Suppose that f has been rotated so that satisfies Note we can write where is a Schwarz function Let . We can write

F U f F f F

(0)a f 0 1a ( )( )

1 ( )a zz z

a z

( ) ic z re ( )

1a cz z

ac

Summary of Campbell’s Proof Let and suppose that so that for some Schwarz Suppose that f has been rotated so that satisfies Note we can write where is a Schwarz function Let . We can write

For we have

F U f F f F

(0)a f 0 1a ( )( )

1 ( )a zz z

a z

( ) ic z re

| | ( )x z m 0 ( )r x m

( )1a cz z

ac

Summary of Proof (Campbell)

Fundamental Inequality [Pommerenke (1964)]

2

2

( ) 1 |1 ( ) | | ( ) | | ( ) | (*)( ) 1 | ( ) | |1 ( ) | | ( ) |

f z x z z z z zF z z z z z z

Summary of Proof (Campbell)

Fundamental Inequality [Pommerenke (1964)]

Two lemmas for estimating

2

2

( ) 1 |1 ( ) | | ( ) | | ( ) | (*)( ) 1 | ( ) | |1 ( ) | | ( ) |

f z x z z z z zF z z z z z z

| ( ) |z

“Small” a Campbell used “Lemma 2” to obtain

where

( ) 1 ( , , ) ( , , 1)( ) 1

f z ba b a k a b k aF z b a b

21 12

xbx

“Small” a Campbell used “Lemma 2” to obtain

where

He showed there is a set R on which k is increasing in a

Let Let

( ) 1 ( , , ) ( , , 1)( ) 1

f z ba b a k a b k aF z b a b

21 12

xbx

1 {( , ) : ( , , 1) 1}C a R k a 1 {( , ) : ( , , 1) 1}A a R k a

“Small” a

“Large” a

Campbell used “Lemma 1” to obtain

where G,C,B are functions of c, x and a

12

2

( ) 1 (1 ) (**)( ) 1

f z G CxF z G B

G H L

“Large” a

Campbell used “Lemma 1” to obtain

where G,C,B are functions of c, x and a

He showed there is a set S on which maximizes at c=r

He showed that (r,x,a) increases on S in a and that

12

2

( ) 1 (1 ) (**)( ) 1

f z G CxF z G B

G H L

( , ,1) 1r x L

“Large” a

Let

Let 2 {( , ) : ( , , ) 0}C a S r x aa

L

2 {( , ) : ( , , ) 0}A a S r x aa

L

“Large” a

Combined Rectangles

Problematic Region

Parameter space below 1.65

Verification of Conjecture

Campbell’s estimates valid in A1 union A2


Find L1 in A1 and L2 in A2


Reduced to verifying Campbell’s conjecture on T

Step 1

Consider the inequality

Show for that

maximizes at

12

2

( ) 1 (1 ) (**)( ) 1

f z G CxF z G B

G H L

2

(1 ) |1 |( , , )|1 ( ) |

x a cG c x aac x a c

16( ( ), ( ), ( ))6 9

G m m l

( , )a T

Step 2


Show at that

is bounded above by

12

2

( ) 1 (1 ) (**)( ) 1

f z G CxF z G B

G H L

66 9

y

11( )1

yg yy

1( ) 1 2.1( 1)(1 )4

l y y

Step 3


Show for that

is bounded above by

12

2

( ) 1 (1 ) (**)( ) 1

f z G CxF z G B

G H L

( , )a T 2 2 2 2 2

222

| 2 | (1 ) ( )(1 )( , , ) (1 )|1 ( ) | (1 ) |1 |

a c ac x x r ac x a xac x a x x a c

H

2 33

4 13 13( ) 1 ( 1) ( 1) ( 1)5 10 10

h

Step 4


Let and

Show that

36( )

6 9g l

3 3( ) ( ) 1g h

2 33

4 13 13( ) 1 ( 1) ( 1) ( 1)5 10 10

h

12

2

( ) 1 (1 ) (**)( ) 1

f z G CxF z G B

G H L

Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells...

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