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Jun 19, 1992 - 1. 1. < qK(r) < th(arth r + (K- 1)/x(r')), for K (1, ), r. (0, 1)with r'= X/1 r2 ... and f(1) 81 1/K. Then fis strictly increasing ifK > 1 and strictly decreasing.
ILLINOIS JOURNAL OF MATHEMATICS Volume 38, Number 3, Fall 1994

FUNCTIONAL INEQUALITIES, JACOBI PRODUCTS, AND QUASICONFORMAL MAPS M.K. VAMANAMURTHY AND M. VUORINEN 1. Introduction

The special function (cf. (2.1))

(1.1)

iz-l( iz( r)/K),

CPK( r )

where K (0, oo), r (0, 1), is closely related to geometric properties of quasiconformal mappings. Some examples of such geometric properties are the quasiconformal Schwarz lemma [LV, p. 64] and the study of the Beurling-Ahlfors extension of quasisymmetric functions [AH], [L], [LV]. We first recall two earlier explicit estimates for the function qr(r) and then give our main results, which yield new identities and inequalities for this frequently occurring function. The basic inequality

rl/K < qK(r) < 41-1/Krl/K

(1.2)

for K (1, oo) and r (0, 1), has been known for more than thirty years. This inequality was recently sharpened [AVV3] to

(1.3)

for K

1

(1, ), r

1.4. THEOREM.

1

(0, 1)with r’=

X/1

r2

(0, oo), let f: [0, 1]

For K

f(r) and f(1)

< qK(r) < th(arth r + (K- 1)/x(r’)),

1

(l/K(r) (l-r) 1/Ic

-

R, be defined by

for O < r
1 and strictly decreasing

Received June 19, 1992. 1991 Mathematics Subject Classification. Primary 30C62; Secondary 33E05. (C) 1994 by the Board of Trustees of the University of Illinois Manufactured in the United States of America

394

FUNCTIONAL INEQUALITIES AND JACOBI PRODUCTS

if K
1, a, b (0, 1) was proved in [AVV1, 3.13]. Our next theorem gives a majorant for the right hand side of (1.6). 1.7. THEOREM.

(1) (2)

(0, 1). Then

Let K >_ 1, a, b

2

oK(ab) < qK(a)PK(b) < qK(V) < OK: (ab), qK( a)qK( b) < qK( al/K)qK( b) 1/K < oK2 (ab),

with equality when K

1.8. THEOREM.

1.

For K > 1, r

(0, 1) the function

f: (0, oo)

--)

( rl/K, 1)

defined by f(p)

(qK(rP)) 1/p,

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M.K. VAMANAMURTHY AND M. VUORINEN

is strictly decreasing. In particular,

r(r



p, p >_ 1, p

11.

Theorems 1.5, 1.7, and 1.8 together give interesting and perhaps new inequalities for infinite products. We shall give various applications of the function q/c(r) to quasiconformal mappings. By the quasiconformal counterpart of the Schwarz lemma [LV, p. 63] the function qr(r)r measures the deviation of a Kquasiconformal automorphism of the unit disk from the identity map. For this function we obtain an explicit majorant in Section 3 and correct an error in an earlier result of the same kind in the literature. The boundary correspondence of a quasiconformal mapping of the upper half-plane onto itself can be characterized as a homeomorphism of the real axis onto itself that satisfies the Beurling-Ahlfors p-condition [BA], [AH]. The p-condition is frequently used in the theory of Teichmiiller spaces [L] and it has been extensively studied in its own right by W.K. Hayman and A. Hinkkanen [HH], [HI]. P. Tukia [T] showed recently that such homeomorphisms, often called quasisymmetric functions, can change the Hausdorff dimension of a set in a very peculiar way. We show here with a quantitative estimate that quasisym1. This result improves the metric functions approach linear maps when p qualitative estimate in [L, p. 32]. See also the interesting recent results of F.P. Gardiner and D.P. Sullivan in [GS]. Our results also complement and improve the earlier growth estimates of these maps in [HH] and [HI]. Our proof makes use of the quasiconformal extension of such a map and the function pr(r). For integer values of K the function pr(r) occurs also in number theory, namely in the study of modular equations [BB, pp. 102-109], [BE1], [BE2] and singular values associated with complete elliptic integrals. Bounds for the function qr(r), such as those in Theorem 1.4, also yield bounds for such singular values. An example is the following corollary to Theorem 1.4 (2). 1.9. COROLLARY. admits the estimate

The pth singular value kp (for definition see 3.18 below)

(1 l/vf) 1/7 < 1 -kp < 8’-’/7(1 1/1/’) 1/v/" for p

1, 2, 3,

Some conjectures concerning the function Section 2.

pr(r) are given at the end of

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FUNCTIONAL INEQUALITIES AND JACOBI PRODUCTS

Acknowledgements. We are indebted to R. Kiihnau for suggesting the question concerning Theorem 1.4 and to G.D. Anderson, J.M. Borwein, and J. Pfaltzgraff for useful remarks and discussions. The research was completed during the first author’s visit to Michigan State University and the second author’s visit to the University of Michigan under a grant from the Academy of Finland and the U.S. National Science Foundation, Grant NSF-DMS9003438 of Prof. F. W. Gehring. 2. Proofs

The notation and terminology will be as in [LV]. The hyperbolic cosine and tangent and their inverse functions are denoted by ch, th, arch, and arth, respectively. The function /x(r), 0 < r < 1, in (1.1) is given by [LV, (2.2), p.

60]

(2.1) /x(r) where

7r

JU’(r) JU(r)

JU(r)

JU’(r)

JU(r’),

dx

f01 V/(1 r’

x2)(1 r2x 2)

/1

r2

We shall need the following differentiation formula from [AWl, 3.27]

(2.2) where s

K

d-

qa/r(r), K (0, oo),

r

r(r’)2jU2(r) (0, 1), r’

V/1

r2

s’

V/1

s2

2.3. Proof of Theorem 1.4. We need only prove the result for K (1, oo), since the other case follows by inversion. Differentiating and using (2.1) and (2.2), we obtain

K. (l

r)l+/IC, f,(r)

(l

s)[l

s(I + s)jU’2(s) r(1 + r)jU’2(r)

]

which is positive since tjU’2(t) is an increasing function of on (0, 1) and 0 < s < r < 1 [AVV2, 2.2(3)]. Finally, it follows from [LV, p. 65] and [AVV1, (3.4)] that limr_ f(r) f(1). r

398

M.K. VAMANAMURTHY AND M. VUORINEN

Theorem 1.4 is related to the conjecture [LV, p. 68] that a K-quasiconformal automorphism of B2 with f(0) 0 satisfies

If(x) f(Y)]

(*)

_
1, r (0, 1) let g be the extremal K-quasiconformal mapping [LV, p. 65] with gB 2 B 2, g(0) 0, and g(r) ql/r(r), g(1) 1. If x r, y 1, then

Ig(x)

g(Y)l

ql/r(r) < 81-1/r11 xl 1/.

1

Hence in this case (.) holds even with a smaller constant. 2.4. Proof of Theorem 1.7. (1) The first inequality is (1.6). To prove the second one we show that for a fixed a (0, 1) the function

f(x)

log qtc(a 2)

+ log qr(x 2)

2log qr(ax)

is increasing on (0, a) and decreasing on (a, 1), so that f(x) x (0, a) td (a, 1). If we write s qr(x2), t qr(ax), u the differentiation formula in [AVV2, Lemma 2.1] we get

f’(x)

< f(a) 0 for all qr(x) and use

2 --(g(x 2) g(ax))

-u2),filfff(u)2/((1 -x2),’-:ff(x)2). Since g(x) is decreasing by [AVV1, Lemma 3.12] it follows that f’(x) > 0 for x (0, a) and f’(x) < 0 for x (a, 1). The third inequality follows from (2).

with g(x)= (1

(2) The proof of part (2) will follow from Theorem 2.22.

-

r3

In connection with the study of quasisymmetric functions of the real line and their extension to quasiconformal mappings of the plane [BA], the function #K

(2.5)

A(K)=

(1) (1)__

(l/g

2

K>O,

has found many applications [L], [LV]. We now consider the following

399

FUNCTIONAL INEQUALITIES AND JACOBI PRODUCTS

generalization of A(K):

2,, IK(X)=A(Kr)=[CPK(r)]

(2.6)

ql/K(r’ )

,

for 0 < K < 0 < r < 1, r’ V/1 r 2 and x (r/r’) 2. S. Agard [A] introduced this function in the study of quasisymmetric functions of the real axis.

2.7. THEOREM. (1) For K

A(K) 7 with equality

(2) For K

1 or r

(0, 1), r’=

V/1

1/x/. 2 r

A(K,r) > K 4 with equality

(3) For K

r2

_ 16 K- ltK,

for all t (0, oo). Proofi Since the cases of equality in (1) and (2) are obvious, we only need to prove the strict inequalities. With s qK(r), using (2.1) and (2.2), we

400

M.K. VAMANAMURTHY AND M. VUORINEN

-

obtain

adr whence

for 1 < K < 1. Hence

,0
(th(K-241/K-1 arth r)) g. qK(r)
u, and

dv dp

v(v’)2d2(v)

Now du

dv

du

Ku( u, ) Z,2 ( u ) dp

sP log s + psp_l

-

u log r,

dS

(

=v logs

+

p ds

7-

)

-

407

FUNCTIONAL INEQUALITIES AND JACOBI PRODUCTS

Hence, log s

(v’)2dU2(v)log r

p ds

+7

Writing m(r)

/z( v)log r

(u,)2dU2(u)K ( v’)2J/(v)J/’( v)log r (2/rr)(r’)2aU(r)aU’(r) this yields

pds s dp

1

1

1

m(u)

so that

p2 ds

1

sdp

m(u)

[m(u)log(1-)-m( v ) log( )] oK(r ) >_ th(d(K)arth(rl/K)),

where

c(K)

max{K, 41-1/K}, d(K)

min{K,41-1/K},

holds, with equality iff K 1 or K 2. The two particular cases K 1 or K 2 are clear with equality in (2.20) for all r (0, 1). Our computational experiments suggest that

(2.21)

q(r)

>_

th(2

2-1/g

arth(A(r)l/K)),

for K > 1 and r (0, 1) where A(r) r/(1 + r’) is as in (2.16). If (2.21) indeed holds, then it is very close to the upper bound in (2.16).

2.22. THEOREM.

For K, L > 1, a, b

(0, 1),

(1) qK(al/L)(qL(b)) 1/K < qK(al/LOL(b) )




1/K- t=

1-1/K)

tl/K(1

>tl/Kt(K-a)/(2K)K--IK log(I)? K K

1 t(K+ 1)/(2K)

log

7

Finally, if 1 < K < (log 8)/2, then h(t) (t3/(4g))log(4/t) is increasing on (0, 1], so that h(t) < log 4, and the result follows. D

In [B, p. 80, Lemma 12], P.P. Belinskii gives the inequality

h(K) < 1 + 12(K- 1)

(3.4)

for K > 1 close to 1. Because the incorrect part of the inequality (3.2) was used in the proof of (3.4) the proof given in [B, pp. 80-82] for (3.4) is not valid. We observe that Corollary 2.11 yields the following improved form of

(3.4). 3.5. COROLLARY.

(3.6)

1

For all K

(1, oo),

+ 7r(K- 1) < A(K) < 1 + a(K- 1)exp(a(K- 1))

4.37688... is as in Corollary 2.11. In particular, then 1/(2a)),

where a

if K (1, 1 +

A(K) < 1 + 8(K- 1).

(3.7) 3.8. THEOREM.

qK(t)

Proof.

For t

(0, 1), t’

X/1

2

and K

(1, oo), we have

< (t’) 2 th((K- 1)/z(t’)) < (K- 1)(t’)210g(4/t’).

The first inequality follows immediately from the upper bound in

(1.3). Since (th x)/x is decreasing on (0, oo), the second inequality is a consequence of the well known property [LV, p. 64, (2.10)] that /z(t’)< log(4/t’). A slightly different final estimate follows if we use the inequalities in Remark 2.24.

3.9. The p-condition. In [BA] Beurling and Ahlfors characterized the boundary correspondence of quasiconformal automorphisms of the upper

FUNCTIONAL INEQUALITIES AND JACOBI PRODUCTS

half plane H 2 as those homeomorphisms

f: R

413

R that satisfy the p-condi-

tion

--1

(3.10)

P


_ 1 [L, p. 34]. For each p _> 1 let us denote by K(p) the smallest constant K such that each homeomorphism f: R R satisfying (3.10) has an extension to a K-quasiconformal mapping of the whole plane R 2 which agrees with f on the real axis. It is well-known by [BA], [AH] and by later results of M. Lehtinen [L, p. 34] that

K(p) < min{2p- 1, p3/2}. It seems to be an open problem whether this inequality is sharp for any p>l. W.K. Hayman and A. Hinkkanen have extensively studied functions satisfying (3.10) independent of quasiconformal extension [HH], [HI]. They obtained sharp bounds for the growth of a function satisfying (3.10) and normalized by the conditions f(0)--0, f(1)= 1. That these conditions are mere normalizations follow from the fact that along with f also h f g satisfies (3.10) with the same p whenever h, g are similarity maps. Alternatively, growth estimates for the functions satisfying (3.10) can also be derived by using quasiconformal extension together with the result of Agard [A] that a K-quasiconformal map f: R z R z satisfies

for all distinct x, y, z follows that

R 2 with

If(x) f(Y)l If(x) f(z)l

-

Ix- y I/Ix- z l. A l/K,

From (3.11) it also

t

3.12. THEOREM. Let f: R R be a homeomorphism satisfying the p-condition (3.10) and let K K(p). Ill(O) 0 and f(1) 1 then for y > 2,

f(y)

Proof

1


dt dK

(r)

Hence by (3.15)

f’(K)