The Verification of an Inequality

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

{ : | | 1}D z z


{ : | | 1}D z z

2

2 | |( ) | |1 | |

dzz dzz

hyperbolic metric


Hyberbolic Geodesics

{ : | | 1}D z z



Hyberbolically Convex Set

{ : | | 1}D z z




Hyberbolically Convex Function

{ : | | 1}D z z




Hyberbolically Convex Function

Hyberbolic Polygono Proper Sides

{ : | | 1}D z z

Examples

2 2

2( )(1 ) (1 ) 4

zk zz z z

k

Examples

12 4 2

0

( ) tan (1 2 cos2 )

2where , 0 2(cos )

z

f z d

K

f

Schwarz NormFor let

and

( )f A D

212f

f fSf f

2 2|| || sup{(1 | | ) | ( ) |: }f D fS z S z z D

|| ||f DS

Extremal Problems for Euclidean Convexity

Nehari (1976):

( ) convex || || 2f Df D S

|| ||f DS


Nehari (1976):

Spherical Convexity Mejía, Pommerenke (2000):



|| ||f DS


Nehari (1976):

Spherical Convexity Mejía, Pommerenke (2000):

Hyperbolic Convexity Mejía, Pommerenke Conjecture (2000):



( ) convex || || 2.3836f Df D S

|| ||f DS

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


Preliminary Facts:

Invariance of hyperbolic convexity under disk automorphisms


Preliminary Facts:

Invariance of hyperbolic convexity under disk automorphisms

Invariance of under disk automorphisms For we have

|| ||f DS, ( )Auto D

1 ( )

|| || || ||

|| || || ||f D f D

f D f D

S S

S S

Classes H and Hn

{ ( ) : ( ) is hyp. convex,(0) 0, (0) 0}

H f A D f Df f

Classes H and Hn

{ ( ) : ( ) is hyp. convex,(0) 0, (0) 0}

H f A D f Df f

{ : ( ) is hyp. polygon}polyH f H f D

Classes H and Hn

{ ( ) : ( ) is hyp. convex,(0) 0, (0) 0}

H f A D f Df f

{ : ( ) is hyp. polygon}polyH f H f D

{ : ( ) has at mostproper sides}

n polyH f H f Dn

Reduction to Hn

Lemma 1. To determine the extremal value of over H, it suffices to determine the value over each Hn. Moreover, each Hn is pre-compact.|| ||f DS

Reduction to Hn

Lemma 1. To determine the extremal value of over H, it suffices to determine the value over each Hn. Moreover, each Hn is pre-compact.

A. Hn is compact

|| ||f DS

{0}

Reduction to Hn


A. Hn is compact B.

|| ||f DS

nn

H H{0}

Reduction to Hn


A. Hn is compact B. C. Schwarz norm is lower semi-continuous

|| ||f DS

nn

H H{0}

Examples

2 2

2( )(1 ) (1 ) 4

zk zz z z

k

Reduction to Re Sf (0)

Lemma 2. For each n > 2,

A.

B. C.

sup || || max | (0) | max Re (0)n nn

f D f ff H f Hf HS S S

Schwarz Norm

For let

and

( )f A D

212f

f fSf f

2 2|| || sup{(1 | | ) | ( ) |: }f D fS z S z z D

|| ||f DS



A. (Nehari)

implies

B. C.

sup || || max | (0) | max Re (0)n nn


2

21 1

11( )2 ( )

n nk k

gk kk k

S zz a z a

2lim 4(Im ) | ( ) | 2 forgz wz S z w U



A. (Nehari)

implies

B. There exist C.

sup || || max | (0) | max Re (0)n nn


2

21 1

11( )2 ( )

n nk k

gk kk k

S zz a z a

for which || || 2n f Df H S




A. (Nehari)

implies

B. There exist C. Invariance under disk automorphisms

sup || || max | (0) | max Re (0)n nn


2

21 1

11( )2 ( )

n nk k

gk kk k

S zz a z a

for which || || 2n f Df H S


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 .


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

and let Ωε be the region bounded by Γε.

{ * ( ) ( ) : }w w w n w w


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

and let Ωε be the region bounded by Γε.

{ * ( ) ( ) : }w w w n w w

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

where

( ) 1( ) ( ) ( )2 1D

zf z zf z f z d oz

( ( )) and| ( ) |

ifd d ef

Two Variations for Hn

Variation #1


Variation #1


Variation #1


Variation #1

Barnarnd & Lewis, Subordination theorems for some classes of starlike functions, Pac. J. Math 56 (1975) 333-366.

*( ) ( ) ( )f z f z o


Variation #2


Variation #2

Schwarzian and Julia Variation Lemma 3. If then

Lemma 4. If and Var. #1 or Var. #2 is applied to a side Γj, then

2 32 3( ) ( )f z z a z a z

23 2(0) 6( )fS a a

2 32 3( ) ( ) nf z z a z a z H

23 3 2

2

2 2

(0) 6 (3 4 )2

(2 2 ) ( )2

j

j

fS a a a d

a a d o

Schwarzian and Julia Variation

In particular,

where

2 23 2

0

6Re (0) Re 3 4 22

Re ( )j

j

f

j

S a a d

K d

2 23 2

6( ) 3 4 22

K a a

Reduction to H2

Step #1. Reduction to H4

Reduction to H2


Step #2. (Step Down Lemma) Reduction to H2

Reduction to H2


Step #2. (Step Down Lemma) Reduction to H2

Step #3. Compute maximum in H2

Reduction to H2 – Step #1

Suppose is extremal and maps D to a region bounded by more than four sides.

nf H


Suppose is extremal and maps D to a region bounded by more than four sides.

Then, pushing Γ5 out using Var. #1, we have

nf H

5

10

Re (0) Re ( ) 0fS K d


Consequently, the image of each side γj under K must intersect imaginary axis


Consequently, the image of each side γj under K must intersect imaginary axis


Suppose is extremal and maps D to a region bounded by exactly four sides.

nf H



nf H



nf H



Then, pushing in the end of Γ3 , near f (z*), using Var. #2, we have

nf H

*

*0

Re (0) ( 1) Re ( ) 0fS K d


Suppose is extremal and maps D to a region bounded by exactly two sides.

nf H

Computation in H2

Functions whose ranges are convex domains bounded by one proper side

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

( )k

( )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

2

0 1sup (1 ) | ( ) |f

xx S x

Computation in H2

Verified A. For each fixed that is

maximized at r = 0

B. The curve is unimodal, i.e., there exists a unique so that

increases for and decreases for At

( )

2(1 ) ( )fr S r

( )

2(0) 2( )fS c

* 0.2182

( )(0)fS

*0 * .2

* ,

*( )2.3836fS

Graph of

*

( )

2(0) 2( )fS c

Documents

The Verification of an Inequality