Majorization-subordination theorems for locally univalent functions. IV A Verification of Campbell’s Conjecture Roger W. Barnard, Kent Pearce Texas Tech University Presentation: May 2008
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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
: Univalent Functions : Convex Univalent Functions
: Linearly Invariant Functions of order U
1 2, K= = U S U
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
Majorization-Subordination Classical Problems (Biernacki, Goluzin, Tao Shah,
Lewandowski, MacGregor)
Let F S
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
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
Majorization-Subordination Campbell (1971, 1973, 1974)
Let F U
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
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
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
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
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
“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
“Large” a
Combined Rectangles
Problematic Region
Parameter space below 1.65
Verification of Conjecture
Campbell’s estimates valid in A1 union A2
Verification of Conjecture
Find L1 in A1 and L2 in A2
Verification of Conjecture
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
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
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
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
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