The Verification of an Inequality
Roger W. Barnard, Kent Pearce, G. Brock Williams
Texas Tech University
Leah Cole
Wayland Baptist University
Presentation: Pondicherry, India
Notation & Definitions

D  {z :| z |  1}
Notation & Definitions

D  {z :| z |  1}
hyperbolic metric
2 | dz |
 ( z ) | dz |
2
1 | z |
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
 Hyberbolically Convex Function
Notation & Definitions

D  {z :| z |  1}
 Hyberbolic Geodesics
 Hyberbolically Convex Set
 Hyberbolically Convex Function
 Hyberbolic Polygon
o Proper Sides
Examples
k ( z ) 
2 z
(1  z )  (1  z )  4 z
2
k
2
Examples

f ( z )  tan  

where  
z
 (1  2
2
1
4  2
cos 2   )
0

2 , 0  
2
K (cos )
f

d 

Schwarz Norm || S f ||D
For f  A( D) let

Sf  

2

f   1  f  
  

f   2 f  
and
|| S f ||D  sup{(1 | z | ) | S f ( z) |: z  D}
2 2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f ( D) convex  || S f ||D  2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f ( D) convex  || S f ||D  2
 Spherical Convexity

Mejía, Pommerenke (2000):
f ( D) convex  || S f ||D  2
Extremal Problems for || S f ||D
 Euclidean Convexity

Nehari (1976):
f ( D) convex  || S f ||D  2
 Spherical Convexity

Mejía, Pommerenke (2000):
f ( D) convex  || S f ||D  2
 Hyperbolic Convexity

Mejía, Pommerenke Conjecture (2000):
f ( D) convex  || S f ||D  2.3836
Verification of M/P Conjecture
 “The Sharp Bound for the Deformation of a Disc
under a Hyperbolically Convex Map,”
Proceedings of London Mathematical Society
(accepted 3 Jan 2006), R.W. Barnard, L. Cole,
K. Pearce, G.B. Williams.
http://www.math.ttu.edu/~pearce/preprint.shtml
Verification of M/P Conjecture
 Preliminary Facts:

Invariance of hyperbolic convexity under disk
automorphisms
Verification of M/P Conjecture
 Preliminary Facts:

Invariance of hyperbolic convexity under disk
automorphisms

Invariance of || S f ||D under disk automorphisms
 For
we have
 ,   Auto( D)
|| S
f
||D  || S f ||D
|| S f ||D  || S f  || 1 ( D )
Classes H and Hn

H  { f  A( D) : f ( D) is hyp. convex,
f (0)  0, f (0)  0}
Classes H and Hn


H  { f  A( D) : f ( D) is hyp. convex,
f (0)  0, f (0)  0}
H
poly
 { f  H : f ( D) is hyp. polygon}
Classes H and Hn

H  { f  A( D) : f ( D) is hyp. convex,
f (0)  0, f (0)  0}
poly
 { f  H : f ( D) is hyp. polygon}

H

H n  { f  H poly : f ( D ) has at most
n proper sides}
Reduction to Hn
 Lemma 1. To determine the extremal value of
|| S f ||D over H, it suffices to determine the value
over each Hn. Moreover, each Hn is pre-compact.
Reduction to Hn
 Lemma 1. To determine the extremal value of
|| S f ||D over H, it suffices to determine the value
over each Hn. Moreover, each Hn is pre-compact.

A. Hn
{0}
is compact
Reduction to Hn
 Lemma 1. To determine the extremal value of
|| S f ||D over H, it suffices to determine the value
over each Hn. Moreover, each Hn is pre-compact.


A. Hn {0} is compact
B. H n  H
n
Reduction to Hn
 Lemma 1. To determine the extremal value of
|| S f ||D over H, it suffices to determine the value
over each Hn. Moreover, each Hn is pre-compact.



A. Hn {0} is compact
B. H n  H
n
C. Schwarz norm is lower semi-continuous
Examples
k ( z ) 
2 z
(1  z )  (1  z )  4 z
2
k
2
Reduction to Re Sf (0)
 Lemma 2. For each n > 2,
sup || S f ||D  max | S f (0) |  max Re S f (0)
f H n

A.

B.
C.

f H n
f H n
Schwarz Norm || S f ||D
For f  A( D) let

Sf  

2

f   1  f  
  

f   2 f  
and
|| S f ||D  sup{(1 | z |2 )2 | S f ( z) |: z  D}
Reduction to Re Sf (0)
 Lemma 2. For each n > 2,
sup || S f ||D  max | S f (0) |  max Re S f (0)
f H n
f H n

A. (Nehari)
f H n
n
k
1 n 1   k2
Sg ( z)  

2
2 k 1 ( z  ak ) k 1 z  ak
implies
lim 4(Im z )2 | S g ( z ) |  2 for w  U
z w


B.
C.
Reduction to Re Sf (0)
 Lemma 2. For each n > 2,
sup || S f ||D  max | S f (0) |  max Re S f (0)
f H n
f H n

A. (Nehari)
f H n
n
k
1 n 1   k2
Sg ( z)  

2
2 k 1 ( z  ak ) k 1 z  ak
implies
lim 4(Im z )2 | S g ( z ) |  2 for w  U
z w


B. There exist f  H n for which || S f ||D  2
C.
Reduction to Re Sf (0)
 Lemma 2. For each n > 2,
sup || S f ||D  max | S f (0) |  max Re S f (0)
f H n
f H n

A. (Nehari)
f H n
n
k
1 n 1   k2
Sg ( z)  

2
2 k 1 ( z  ak ) k 1 z  ak
implies
lim 4(Im z )2 | S g ( z ) |  2 for w  U
z w


B. There exist f  H n for which || S f ||D  2
C. Invariance under disk automorphisms
Julia Variation
 Let Ω be a region bounded by a piece-wise analytic curve
Γ and φ(w) piece-wise C1 on Γ .
Julia Variation (cont)
 Let Ω be a region bounded by a piece-wise analytic curve
Γ and φ(w) piece-wise C1 on Γ .
Julia Variation (cont)
 Let Ω be a region bounded by a piece-wise analytic curve
Γ and φ(w) piece-wise C1 on Γ . At each point w on Γ
(where Γ is smooth), let n(w) denote the unit outward
normal to Γ. For small ε let
  {w *  w   (w)n( w) : w }
and let Ωε be the region bounded by Γε.
Julia Variation (cont)
 Let Ω be a region bounded by a piece-wise analytic curve
Γ and φ(w) piece-wise C1 on Γ . At each point w on Γ
(where Γ is smooth), let n(w) denote the unit outward
normal to Γ. For small ε let
  {w *  w   (w)n( w) : w }
and let Ωε be the region bounded by Γε.
Julia Variation (cont)
 Theorem. Let f be a conformal map from D on Ω
with f (0) = 0 and suppose f has a continuous
extension to ∂D. Then, for sufficiently small ε the
map fε from D on Ωε with fε (0) = 0 is given by
 zf ( z ) 1   z
f ( z )  f ( z ) 
d   o( )

2 D 1   z
where
d 
 ( f ( ))
d and   ei
| f ( ) |
Two Variations for Hn
 Variation #1
Two Variations for Hn
 Variation #1
Two Variations for Hn
 Variation #1
Two Variations for Hn
 Variation #1
f ( z )  f ( z )  o( )
*
 Barnarnd & Lewis, Subordination theorems for
some classes of starlike functions, Pac. J. Math 56
(1975) 333-366.
Two Variations for Hn
 Variation #2
Two Variations for Hn
 Variation #2
Schwarzian and Julia Variation
 Lemma 3. If
f ( z )   ( z  a2 z 2  a3 z 3 
) then
S f (0)  6(a3  a22 )
 Lemma 4. If f ( z )   ( z  a2 z 2  a3 z 3  )  H n and Var. #1 or
Var. #2 is applied to a side Γj, then



S f (0)  6  a3 

2




  a2 

2


 (3a3  4a2   )d  
j

2

 (2a2  2 )d  
j

2

  o( )


Schwarzian and Julia Variation
 In particular,

6
2
2
Re S f (0)
   Re
3
a

4
a

2

d


3
2

2
 0
j
  Re K ( j )  d 
j
where
6
2
2
K ( ) 
3
a

4
a

2



3
2
2
Reduction to H2
 Step #1. Reduction to H4
Reduction to H2
 Step #1. Reduction to H4
 Step #2. (Step Down Lemma) Reduction to H2
Reduction to H2
 Step #1. Reduction to H4
 Step #2. (Step Down Lemma) Reduction to H2
 Step #3. Compute maximum in H2
Reduction to H2 – Step #1
 Suppose f  H n is extremal and maps D to a
region bounded by more than four sides.
Reduction to H2 – Step #1
 Suppose f  H n is extremal and maps D to a
region bounded by more than four sides.
 Then, pushing Γ5 out using Var. #1, we have

Re S f (0)
 Re K ( 1 )  d   0

 0
5
Reduction to H2 – Step #1
 Consequently, the image of each side γj under K
must intersect imaginary axis
Reduction to H2 – Step #1
 Consequently, the image of each side γj under K
must intersect imaginary axis
Reduction to H2 – Step #2
 Suppose f  H n is extremal and maps D to a
region bounded by exactly four sides.
Reduction to H2 – Step #2
 Suppose f  H n is extremal and maps D to a
region bounded by exactly four sides.
Reduction to H2 – Step #2
 Suppose f  H n is extremal and maps D to a
region bounded by exactly four sides.
Reduction to H2 – Step #2
 Suppose f  H n is extremal and maps D to a
region bounded by exactly four sides.
 Then, pushing in the end of Γ3 , near f (z*), using
Var. #2, we have

Re S f (0)
 (1) Re K ( * )  d   0

 0
*
Reduction to H2 – Step #2
 Suppose f  H n is extremal and maps D to a
region bounded by exactly two sides.
Computation in H2
 Functions whose ranges are convex domains
bounded by one proper side (k )
 Functions whose ranges are convex domains
bounded by two proper sides which intersect
 Functions whose ranges are odd symmetric
convex domains whose proper sides do not
intersect ( f )
Computation in H2
 Using an extensive calculus argument which
considers several cases (various interval ranges for
|z|, arg z, and α) and uses properties of
polynomials and K, one can show that this
problem can be reduced to computing
sup (1  x) 2 | S f ( x) |
0 x 1
Computation in H2
 Verified
(1  r )2 S f ( ) (r )

A. For each fixed  that
maximized at r = 0

B. The curve S f  (0)  2(c   2 ) is unimodal,
i.e., there exists a unique  *  0.2182 so that
S f (0) increases for 0     * and
decreases for  *     2 . At  * ,
( )
 ( )
Sf
 ( * )
 2.3836
is
Graph of S f
 ( )
*
(0)  2(c   )
2