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Transcript of Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells...
![Page 1: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/1.jpg)
Majorization-subordination theorems for locally univalent functions. IV
A Verification of Campbell’s Conjecture
Roger W. Barnard, Kent PearceTexas Tech University
Presentation: May 2008
![Page 2: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/2.jpg)
Notation
{ : | | 1}z z D
![Page 3: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/3.jpg)
Notation
( )DA
{ : | | 1}z z D
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Notation
Schwarz Function
( )DA
: , | ( ) | | |z z on D D D
{ : | | 1}z z D
( ) DA
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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
![Page 6: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/6.jpg)
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
![Page 7: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/7.jpg)
Notation
: Univalent Functions : Convex Univalent Functions
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Notation
: Univalent Functions : Convex Univalent Functions
: Linearly Invariant Functions of order U
1 2, K= = U S U
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Notation
: Univalent Functions : Convex Univalent Functions
: Linearly Invariant Functions of order
Footnote: , and are normalized by
U
1 2, K= = U S U
U 22( )f z z a z
![Page 10: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/10.jpg)
Majorization-Subordination Classical Problems (Biernacki, Goluzin, Tao Shah,
Lewandowski, MacGregor)
Let F S
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Majorization-Subordination Classical Problems (Biernacki, Goluzin, Tao Shah,
Lewandowski, MacGregor)
Let
A. | |If f F on find r so that f F on z r D
F S
(1967) : 2 3M r
![Page 12: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/12.jpg)
Majorization-Subordination Classical Problems (Biernacki, Goluzin, Tao Shah,
Lewandowski, MacGregor)
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
![Page 13: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/13.jpg)
Majorization-Subordination Campbell (1971, 1973, 1974)
Let F U
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Majorization-Subordination Campbell (1971, 1973, 1974)
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
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Majorization-Subordination Campbell (1971, 1973, 1974)
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
![Page 16: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/16.jpg)
Campbell’s Conjecture Let
2
, | | ( )
( ) 1 2 1 1.65
If f F on then f F on z m
where m for
D
F U
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Campbell’s Conjecture Let
Footnote: Barnard, Kellogg (1984) verified Campbell’s for
2
, | | ( )
( ) 1 2 1 1.65
If f F on then f F on z m
where m for
D
F U
1K= = U
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Summary of Campbell’s Proof Let and suppose that so that for some Schwarz
F U f F f F
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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
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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
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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
![Page 22: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/22.jpg)
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
![Page 23: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/23.jpg)
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
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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
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“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
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“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
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“Small” a
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“Small” a
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“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
![Page 30: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/30.jpg)
“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
![Page 31: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/31.jpg)
“Large” a
Let
Let 2 {( , ) : ( , , ) 0}C a S r x aa
L
2 {( , ) : ( , , ) 0}A a S r x aa
L
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“Large” a
![Page 33: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/33.jpg)
“Large” a
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Combined Rectangles
![Page 35: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/35.jpg)
Problematic Region
Parameter space below 1.65
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Verification of Conjecture
Campbell’s estimates valid in A1 union A2
![Page 37: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/37.jpg)
Verification of Conjecture
Find L1 in A1 and L2 in A2
![Page 38: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/38.jpg)
Verification of Conjecture
Reduced to verifying Campbell’s conjecture on T
![Page 39: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/39.jpg)
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
![Page 40: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/40.jpg)
Step 2
Consider the inequality
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
![Page 41: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/41.jpg)
Step 3
Consider the inequality
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
![Page 42: Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbells Conjecture Roger W. Barnard, Kent Pearce Texas Tech.](https://reader031.fdocuments.us/reader031/viewer/2022011801/5a4d1bab7f8b9ab0599cae44/html5/thumbnails/42.jpg)
Step 4
Consider the inequality
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