A geometric approach to two-point comparisons for hyperbolic and Euclidean geometry

A GEOMETRIC APPROACH TO TWO-POINT COMPARISONS FOR HYPERBOLIC AND EUCLIDEAN

GEOMETRY

SEONG-A KIM AND DAVID MINDA

ABSTRACT. Two-point comparison theorems between hyperbolic and euclidean geometry for convex regions in the complex plane <C are known ([5], [6]). We give new geometric proofs of sharp two-point comparison theorems for convex regions.

1. Introduction

Sharp two-point comparison theorems between hyperbolic and eu- clidean geometry are known for various types of regions in the complex plane C ([5], [6], [7], [8]). These comparison theorems were motivated by work of Blatter [1] dealing with a characterization of univalent functions.

The proofs of these results rely upon coefficient estimates for certain classes of functions defined on the unit disk IDJ. The purpose of this note is to present new geometric proofs of two-point comparison theorems for convex regions. Our proofs show that these comparisons can be derived from the fact that the reciprocal of the density of the hyperbolic metric is a concave function on convex regions [10]. This concavity property actu- ally characterizes convex regions [4]. Other characterizations of convex regions in terms of hyperbolic geometry are given in [3].

We recall the known comparisons for convex regions. We begin with a brief discussion of hyperbolic geometry. Suppose n is a convex region in C with n

c. Let An(w)ldwl denote the hyperbolic metric on n. For the unit disk A])(z)ldzl = Idzl/(l -lzI

The density An is determined from An(f(z)) = 1j[(1-lzI

)1f'(z)l]' where f is any conformal mapping

Received June 1, 1999.

1991 Mathematics Subject Classification: 30C45.

Key words and phrases: the hyperbolic metric, convex region.

This research was supported in part by a grant from the Korean Council for University Education, 1998.

of the unit disk onto n. The hyperbolic distance between A, BEn is given by

dn(A,B) = inf l ^An(w)ldwl,

where the infimum is taken over all rectifiable paths

in n joining A and B. A path 8 connecting A and B is called a hyperbolic geodesic if

dn(A, B) = 1 ^An(w)ldwl·

For the unit disk

I ^a-b I

dlIJ)(a, b) = artanh 1- ab

and hyperbolic geodesics are arcs of circles that are orthogonal to the unit circle aID. Hyperbolic geodesics exist in n and are the images of hyperbolic geodesics in

under a conformal map f : ID

n.

In addition to supporting hyperbolic geometry, n carries the euclidean geometry that it inherits as a subset of C. Two-point comparison theo- rems bound the euclidean distance lA - BI above and below in terms of the hyperbolic distance dn(A, B) and the values An(A) and An(B). For a convex region n and any

2: 1 the lower bound

sinh(dn(A, B)) [ 1 1 ]

A B

[2cosh(Pdn(A,B))]1/p An(A)p + An(B)p ~ I - I

and the upper bound

lA - BI < [2cosh(pdn(A,B))Jl/psinh(dn(A,B)) - [An(A)p + An(B)p}1/p

are known ([5] and [6]) .. Both bounds are sharp: equality holds in either for distinct A, BEn if and only if n is a half-plane and the euclidean line through A and B is perpendicular to the edge of the half-plane.

The lower (upper) bound is a nonincreasing (riondecreasing) function of p 2: 1. Therefore, the bounds for p = 1, namely,

sinh(dn(A, B)) [ 1 1 ]

(1) 2 cosh(dn(A, Bn An(A) + An(B) ~ ^{lA - BI}

< 2cosh(dn(A, B)) sinh(dn(A, B))

- An(A) + An(B)

are the strongest. The limiting cases

=

are the weakest:

sinh(dn(A, B)) < lA _ BI

exp(dn(A, B)) min{>.n(A), An(B)} -

exp(dn(A, B)) sinh(dn(A, B))

< .

- max{An(A), An(B)}

These weakest comparisons are invariant versions of the classical growth theorem

Izl < I (z)1 < Izl

1 + Izl - 9 - 1 - Izl

for normalized (9(0)

0,9'(0)

1) convex univalent functions 9 de- fined on]])l. We shall provide a new, geometric proof of the strongest inequalities (1).

2. Preliminaries

Two quantities associated with the hyperbolic metric play an impor- tant role in our work. These quantities are the connection

fn(w) = 2aIOgAn(w)

aw

and the Schwarzian

Sn(w) = 2 [0' _~;''''(w) ^_ (81;;: ^{An (w))']}

of!] 1

ow (w) - 2 fn (w) . Note that

of!] 1

ow (w) = Sn(w) + 2 fn (w) . Also,

ofn (w) = 2An(W)2

&w

follows from the fact that An(w)ldwl has curvature -4; that is,

~ ^{log An (w) 4}

a:~~n ( ^w)

-4= An(W)2 An(w)2 .

IBn(w)1 2An(w)2'

Various characterizations of convexity are conveniently expressed in terms of the connection and the Schwarzian.

PROPOSITION

1. Suppose

is a hyperbolic region in C. Then the following are equivalent.

(i)

is convex.

(ii)

is concave on

(iii) IBn(w)1 +

2An(W)2 for wE

(iv) Irn(w)1

2An(w) for wE

The equivalence of (i), (ii) and (iii) is established in [4], but in different notation. The simple identities

8

(...!...)

1

An(W) 8w2 (w)

Irn(w)1

= _{An(W) ,}

show that Theorem 1 of [4] contains the equivalence of (i), (ii) and (iii).

Note that (iii) implies (iv). The equivalence of (i) and (iv) is given in [2]. See [9] for a geometric proof that (i) implies (iv), as well as for a proof that equality holds in (iv) if and only if

is a half-plane. In other words, Irnl- 2An when

is a half-plane.

In addition to these characterizations of convex regions we require an elementary result for a differential inequality.

PROPOSITION

2. Suppose u, v

C2[- L, L], v"

4v and u" = 4u. If u(L) = v(L) and u( -L) = v( -L), then either v = u on [-L, L] orv > u on (-L,L).

For a proof of this result see [6J.

3. Main result

THEOREM

1. Suppose

is a convex region in C with

-=I- C.

(i) For A,B

sinh(dn(A, B)) [1 1]

(2) 2cosh(dn(A, B)) An(A) + An(B) ~ ^{lA -} El·

(ii) For A,B

f2

(3) ^IA-BI < 2cosh(dn(A,B)) sinh(dn(A, B)).

- An(A) + An(B)

Equality holds in (2) or (3) for distinct A and B if and only if n ^{is a}

half-plane and the euc1idean line through A and B is perpendicular to the edge off2.

Proof. (i) Fix A, B E f2 with A

B. Because f2 is convex, the straight line segment /

[A,B] is contained in f2. Let / : w

w(s), -L ::; s ::; L, be a hyperbolic arclength parametrization of /. This means that w'(s) = e

jAn(w(s)), where e ^is the argument of B-A, and 2L is the hyperbolic length of /. Set

1

v(s)

An(w(s)) ' -L::; s ::; L.

Then

v'(s) ^{2 Re} {OAn(W(s)) W'(S)}

An(W(S))2 OW

1 { iIJ}

= -

An(W(S)) Re fn(w(s))e -v

(s)Re {fn(w(s))e

Next, we compute the second derivative of v.

v"(s) -2v(s)v'(s)Re {fn(w(s))e

v

(s)Re {e

~ ^{f(w(s)) }}

2v

(s)Re

{fn(w(s))e

v

(s)Re {e

~ fn(w(s)) } . Now,

ds fn(w(s)) d of n of n - -

~(w(s))w'(s)

+

[ uw uw]

1

Sn(w(s)) + "2fn(w(s)) An(W(S}) + 2An(w(s))e- ,

so that

v"(s) - 2v

(s)Re 2{rn(w(s))eiO } v 2

(s) { [ ^{1 (} 2] ^2iO}

- An(W(S)) Re Sn(w(s)) + 2fn w(s)) e -2v2(s )An( w(s))

= v

(s) [2Re 2 {fn(w(s))eiO }

-Re { [Sn(w(s)) + ~fn(w(s))2] ^{e 2iO }] - 2v(s)}

_ v

(s) [lfn(w(s))1

-Re {[Sn(w(s)) - ~fn(w(s))2] ^{e 2iO }] - 2v(s)}

since

Because n is convex we obtain

v"(s) < v

(s) [Irn(w(s))1 2 + ISn(w(s)) - ~fn(w(s))21] ^{- 2v(s)}

< v

(s) [lrn(w(s))1 2

+ ^ISn(w(s))1 + ~ ^{Ifn(w(s))1 2 ] - 2v(s)}

< v

(s) [(2An(w(s)))2 + 2An(w(s))2] - 2v(s)

= 4v(s).

Note that if v"(s) = 4v(s) , then Ifn(w(s))1 = 2An(w(s)), and so n must be a half-plane.

Let u(s) = ccosh(2s) + dsinh(2s) be the solution of u"(s) = 4u(s) . that satisfies the boundary conditions u( -L)

v(-L) andu(L) =v(L).

Then

c d

1 1

v(L)+v(-L)

2 cosh(2L)

2 cosh(2L) , veL) - v(-L)

2sinh(2L) .

Proposition 2 implies that either v = u on [-L,L] or v > u on (-L,L).

Now,

lA - El

!Idw

1 _-L

^Iw'(s)lds

-I: ^{An(~(s)) =} I: ^v(s)ds

> 1: ^u(s)ds

^csinh(2L)

sinh(2L) [ 1 1 ]

2cosh(2L) An(A) + An(B) - ~ ^tanh(2L) [An~A) ⁺ An~B)] ^.

Note that equality implies v = u and so v"(s) = 4v(s) on [-L, L]; this implies 0 is a half-plane. Since 2L ;::: dn(A, B) with equality if and only if 'Y is a hyperbolic geodesic, we conclude that

1 [ 1 1 ]

lA - BI > "2 tanh(dn(A, B)) An(A) + An(B)

= sinh(dn(A, B)) [ 1 1 ]

2cosh(dn(A, B)) An(A) + ^{An(B) .}

Equality implies that 0 is a half-plane and the straight line segment 'Y

[A, Bl is a hyperbolic geodesic. In a half-plane the hyperbolic geodesics are arcs of circles orthogonal to the edge of the half-plane and segments of lines perpendicular to the edge of the half-plane. Hence, if equality holds, then 0 is a half-plane and [A, B)

part of a line perpendicular to the edge of the half-plane.

All that remains is to show that equality holds when 0 is a half-plane and [A, B] is perpendicular to aO. It suffices to consider the particular half-plane JHI = {w : Im{w} > O}. Then AJ8[(W) = 1/[2Im{w}]. Fix u

lR and 0 < a < b. Then for A

u+ia and B

u+ib, IA-BI

b-a, dJ8[(A, B) = pog~, and AJ8[(A) = 1/(2a), AJ8[(B) = 1/(2b). It is now straightforward to check that equality holds in (2).

(ii) Consider A, B E 0 with A

B. Let 8 be the hyperbolic geodesic

joining A to B. Suppose 8 : w

w(s), -L ::; s ::; L, is a hyperbolic

arclength parametrization of 8. Then 2L

dn(A, B), the hyperbolic

length of 8, and w'(s)

An(W(S)), where

is a unit tangent to

oat w(s). Set

Then

V(s)

An(w(s)).

Kh(W(S),O)

V'(S) 2Re {OZ(W(S»W'(s)}

{

oAn ei8 (S)}

2Re Ow (w(s) An(w(s»

Re {rn(w(s»eill(S)}.

Because n ^is convex we obtain

(4) 1V'(s) I

Irn(w(s»1

2An(w(s»

2V(s).

Next, we calculate the second derivative of V.

V"(s) Re {ei8(S) ~ ^{rn(w(s» } +} Re { rn(w(s» ~ ^ei8(S)}

An(~(s»Re ^{{[Sn(w(s» +} ~rn(w(s)?] ^{e 2ill (S)}}

+2An(w(s» + di:)Re {ieill(s)rn(w(s»)}.

The hyperbolic curvature of 0 is

Ke(W(S),o) + 2Im {~(w(s»ei8(s)}

An(w(s»

Ke(W(S),o) + ^III {rn(w(s»ei8(s)}

An(w(s»

where

1 {w"(s)}

Ke(W(S), 0)

IW'(s)IIm w'(s).

is the euclidean curvature of'Y at w( s). Since 0 is a hyperbolic geodesic, Kh(W(S),o) =0, so

Im {rn(w(s))ei8(s)} = -Ke(W(S), 0).

From w'(s)

eill(s) jAn(w(s)), we obtain Im {w"(s)}

dOes)

w'(s) ds

D

and so

dO(s)

Ke

(w(s),8) =

Thus,

dO(s) = _ 1 I {r ( ()) ill(s)}

ds AI1(W(S))

w s e

and so

V"(s) = AI1(~(s))Re {[SI1(w(S)) + ~rl1(w(s))2] ^{e 2i11 (S)}}

+2AI1(w(s)) + AI1(~(S)) ^Im

{rl1 (w(s))e ill (s)}

since

Finally,

V"(s) = AI1(~(S)) (~lrl1(w(s))12 ^{+ Re} {SI1(w(s))e 2iO (s)} ) + 2AI1(w(s)), since

Because n is convex, we get

V"(s) ::; AI1(~(S)) (~lrl1(w(s))12 ⁺ ISI1(w(s))I) + 2An(w(s)) ::; 4AI1(w(s))

4V(s).

Now, let U(s) = Ccosh(2s) + Dsinh(2s) be the solution of U"(s)

4U(s) that satisfies the boundary conditions U(- L )

V (- L) and U(L) = V(L). Then

C = V(L) + V( -L) = AI1(A) + AI1(B) > 0 2cosh(2L) 2cosh(dl1(A, B)) , V(L) - V(-L)

2sinh(2L) .

Proposition 2 implies V = U on [-L, L] or V > U on (-L, L). Now, if U > 0 on [-L,L]

lA - BI < lldwl _a ⁼ 1

^Iw'(s)lds 1

^ds 1

^ds

-L An(W(S)) = -L V(s)

< 1

_Ccosh(2s) ^ds _{+ Dsinh(2s)'}

with equality if and only if the hyperbolic geodesic

is the euclidean line segment [A, B]. We show that U > 0 on [-L, L]. Note that

1 C cosh(2s) - D sinh(2s) U(s) = C2 cosh2(2s) - D2 ^sinh

^(2s)

1 cosh(2s) - 7sinh(2s) - Cl + (1- 7

sinh2(2s)' where

D V(L) - V(-L) 1

7

C

Le'-)

_-L::7-) tanh(2L)

We will show that 171 :::; 1; in particular, this shows that U > 0 on [-L, L]. From (4) we have

-2 < V'(s) < 2 - V(s) -

for s

L, L]. If we integrate these inequalities over the interval [-L,L]' then we obtain

V(L)

-4L :::; log V( -L) :::; 4L, or

e-4L < V(L) <e4L . - V(-L) -

The function h(t) = (t-1)j(t+ 1) is increasing for t > -1 since h'(t) =

2j(t + 1)2 > O. Thus,

h(e-4L ) < h ( V(L) ) < h(e4L)

- V(-L)

,

or

l

^{Ccosh(2s) ds + Dsinh(2s)}

V(L) - V(-L)

- tanh(2L):::; V(L) + V(L) :::; tanh(2L).

This demonstrates that 171 :::; 1. Also, we conclude that 7 = ±1 implies JV'(s) I = 2JV(s)l, or Ifn(w(s))1 = 2An(w(s)), which means n is a half- plane. Now

~ 1

^{cosh(2s) -}

^{sinh(2s) ds}

C

1 + (1 -

sinh

(2s)

=

~ 1

^{cosh(2s) ds}

C

1 + (1 -

sinh

(2s) since sinh(2s)/[1 + (1 -

sinh

(2s)] is an odd function. Also,

j

---'---",---- ds ^cosh(2s) < 1

cosh(2s) ds

-L

1 + (1 -

sinh

(2s

= sinh(2L) - sinh(dn(A, B)),

and equality implies

= ±l. By combining our inequalities, we obtain lA _ BI < sinh(dn(A, B)) = 2cosh(dn(A, B)) sinh(dn(A, B))

- C An(A) + An(B)

and equality implies n is a half-plane and 6 = [A, B] is a hyperbolic geodesic, so [A, B] must be perpendicular to an.

Conversely, it is routine to show that if n is a half-plane and [A, B]

lies on a line orthogonal to an, then equality holds in (3).

4. Acknowledgement

Both authors want to thank Professor Kang Tae Kim for the kind invitation to visit POSTECH May 2-3, 1997. This work was begun during the visit. The second author thanks Professor Jae Keol Park of Pusan National University for arranging his visit to Korea in May, 1997.

Also, the first author thanks Department of Mathematics of POSTECH

for their allowing her visit to the Department during the 1998 academic

year.

References

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Helv.

[2]

Harmelin, Hyperbolic metric, curvature of geodesics and hyperbolic discs in hyperbolic plane domains, Israel J. Math.

[3]

S. Kim, On convex hyperbolic regions, Japanese J. Math

[4] S. Kim and D. Minda, The hyperbolic and qv.asihyperbolic tnetrics in convex regions, J. Analysis 1

[51 - - , Two-point distortion theorems for univalent functions,

J.

Math.

[6] W. Ma and D. Minda, Two-point distortion theorems for strongly close-to- convex functions, Complex Variables Theory Appl. 33·

[7] _ _ , Two-point distortion for univalent functions, J. Comp. Appl. Math., to appear.

[8] F. Maitani, W. Ma and D. Minda, Two-point comparisons between hyperbolic and euclidean geometry on plane regions, submitted.

[9] D. Minda, Applications of hyperbolic convexity to euclidean and spherical convexity, J. Analyse Math.

[101 D. Minda and D. Wright, Univalence criteria and the hyperbolic metric,

Mtn. J. Math.

Department of Mathematics Woosuk University, Wanju-gun Cheonbuk 565-701, Korea

E-mail: [email protected] Department of Mathematical Sciences University of Cincinnati

Cincinnati, OH 45221-0025, USA

E-mail: [email protected]