Bounds for Quasiconformal Distortion Functions

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 05, ARTICLE NO AY96505 Bounds for Quasiconformal Distortion Functions S L Qiu School of Science and Arts, Hangzhou Institute of Electronics Engineering, Hangzhou , People s Republic of China M Vamanamurthy* Department of Mathematics, Uniersity of Auckland, Auckland, New Zealand and M Vuorinen ( ) Department of Mathematics, P O Box 4 Yliopistonkatu 5, FIN-00014, Uniersity of Helsinki, Helsinki, Finland Submitted by J L Brenner Received May 3, 1994 Several new inequalities are proved for the distortion function Ž r appearing in the quasiconformal Schwarz lemma Other related special functions are studied and applications are given to quasiconformal maps in the plane Some open problems are solved, too 1997 Academic Press 1 INTRODUCTION The so-called quasiconformal -distortion function is defined as / 1 1 Ž r ž Ž r, 0r1, 0, Ž 11 * address: vamanamu@mathaucklandacnz address: vuorinen@cscfi X97 $500 Copyright 1997 by Academic Press All rights of reproduction in any form reserved

2 44 QIU, VAMANAMURTHY, AND VUORINEN where Ž r Ž r Ž 1 Ž r and BF, WW d Ž r H, Ž r Ž r, 0 1r sin r 1 r As suggested by its name, this function has found important applications in the theory of quasiconformal mappings LV, P, AVV4, Q, and it has been the subject matter of the recent papers AVV1, AVV, VV In VV, V, it is also pointed out that the function Ž r occurs in number theory, in particular, in Ramanujan s work on modular equations and singular values of elliptic integrals B, BB, V Some of the function-theoretic applications rely on explicit estimates for the rather involved function Ž r An example of such an explicit inequality is LV, He r 1 r 4 11 r 1, 13 for all 1 and r Ž 0, 1 Some other special functions of the geometric theory of quasiconformal maps can be expressed in terms of Ž r For example, ž ( / ž / r 1 Ž r 1, r, Ž 0,, Ž 14 1r 1r is a function occurring in the study of both quasisymmetric functions and quasiconformal mappings A, LV The goal of this paper is to sharpen some existing inequalities and to derive concavity and convexity results for the functions Ž r and Ž r from which new functional inequalities follow We now state some of our main results 15 THEOREM For each r Ž 0, 1, define fž Ž rž 1 r r 1 and gž Ž r1r r Then fž and gž 1 are strictly decreasing and increasing on 1,, respectiely, with f 1 and g In particular, 11 1 Ž r Ž 1r r Ž 1

3 QUASICONFORMAL DISTORTION FUNCTIONS 45 and for all 1 and r 0, r 1r r 16 THEOREM For each 1, the function fž x arth Ž th x is strictly increasing and concae from Ž 0, onto Ž 0, In particular, for a, bž 0, 1, ž / a b Ž a Ž b, Ž 1 1ab ab 1 Ž a Ž b Ž a Ž b with equality if and only if 1 or a b, ž / a b Ž a Ž b 1ab 1 Ž a Ž b 1 1 Ž Ž ab Ž 1ab ab, Ž 1 Ž Ž ab Ž 1ab ab with equality in the first if and only if 1 and in the second if and only if 1 or a b 17 THEOREM For 1, let f and g be defined on R by fž x log Ž x Ž Ž x e and g x e 1 Then f is strictly increasing and conex with fž R R and 1 fž x for all x Ž,, while g is strictly decreasing and conex with gž R Ž 0, In particular, 4 a b ž Ž a Ž b / Ž ab Ž a Ž b, Ž 1 with equality in the first if and only if a b and in the second if and only if 1 or a b, and 1 b Ž b b ž a/ a ž a/ for 0 a b, with equality if and only if 1 or a b In the sequel we shall prove several further inequalities of this kind, solve some open problems on Ž r, and use these inequalities and solutions to sharpen known distortion results for plane quasiconformal mappings Our notation is fairly standard, as in LV, except for our

4 46 QIU, VAMANAMURTHY, AND VUORINEN abbreviation for hyperbolic functions We let th denote the hyperbolic tangent function and let arth denote its inverse Whenever x is in 0, 1, we let x 1 x PRELIMINARY RESULTS In this section, we establish some lemmas which are needed in the proofs of our main results 1 LEMMA There exists a unique r Ž 0, 1 0 such that the function FŽ r Ž1r Ž r Ž r log r logž 1 r is strictly decreasing on Ž0, r and strictly increasing on r, 1 In particular, Ž 1r Ž r Ž r log r logž 1 r for each r 0, 1 The first equality holds if and only if r 1, the second equality holds if and only if r 0 or r 1 Proof By differentiating and using Legendre s relation BF, 11010, we get df rr rž r EŽ r F1Ž r, 3 4 dr where H E r 1r sin d and E r E r 4 0 are the complete elliptic integrals of the second kind Then r df 1 FŽ r, 5 r dr where 4 4 F r r r r E r r E r E r r r By AVV3, Theorem ŽŽ 3, 7, it follows that F is strictly decreasing from Ž 0, 1 onto Ž, Since F Ž 0 F Ž , the equation F Ž r 0, for r Ž 0, 1, has a unique solution r Ž 0, Moreover, F Ž r 0if r0, Ž r, and F Ž r 0if rž r, Hence F is strictly decreasing on Ž0, r and strictly increasing on r, 1 0 0

5 QUASICONFORMAL DISTORTION FUNCTIONS 47 Since FŽ0 FŽ1 0, the second inequality in follows The first inequality in appears in W It is also implied by AVV3, Lemma 4Ž 1 6 Remarks Ž 1 The second inequality in is a significant improvement of the previously known inequality W; AVV3, Lemma 4Ž 1 r Ž r Ž r log r log 4 Ž Unfortunately, the function FŽ r in Lemma 1 is not globally convex on Ž 0, 1 In fact, we have d F 3 r r F3Ž r, 4 dr where F r r r r E r r Ž r EŽ rež r 3 r Ž r4 Clearly, F Ž r tends to as r1, so that F Ž r 3 tends to as r tends to 1 On the other hand, we can show that F Ž r tends to as r tends to 0 In fact, EŽ r EŽ r r Ž r FŽ r 3 4 1rr r r 1 Ž r EŽ r r Ž r r r Letting r 0, we see that on the left side, by AVV3, Theorem Ž 7, the second and third expressions have finite limits Ž 4 Ž 4 0 By l Hopital s ˆ Rule, the right side tends to as r 0 Hence F Ž r as r 0 Thus F is concave near 0 and convex near 1 Now, we recall that He; AVV3, Lemma 1 dž r 1, 7 dr 4 rr Ž r, 8 r rr Ž r s ss s

6 48 QIU, VAMANAMURTHY, AND VUORINEN and s ss Ž s Ž s, 9 where s Ž r,0r1, 0 10 LEMMA The functions Ž r Ž r E Ž r E Ž r r Ž r 4 fž r and gž r logž 1r r r are strictly increasing from Ž 0, 1 onto Ž 8, 1 Ž and 3, 1, respectiely Proof Let FŽ r be the numerator in fž r Then r FŽ r E Ž r r Ž r 3 r r so that the result for f follows from AVV3, Theorem Ž 7 ; AVV5, Lemma Next, let Gr be the numerator in gž r Then GŽ r Ž r E Ž r, 3 4 4r r so that the result for g follows from AVV6, Theorem 1Ž 6 ; AVV5, Lemma 11 LEMMA For 1, r Ž 0, 1, let s Ž r Then the functions fž r Ž sr Ž s Ž r, gž r Ž sr Ž s Ž r, hž r Ž sr Ž s Ž r, and kž r Ž sr Ž s Ž r are all strictly de- Ž creasing on 0, 1, with ranges 0, 1, 1,,,, and Ž0, 1, respectiely Proof By differentiation and simplification, we see that fž r 0, if and only if FŽ s FŽ r, where FŽ x Ž x Ž x Ž x EŽ x But this inequality is true by Lemma 10 and AVV3, Theorem Ž 3 The proof for gž r is similar Next, since hr g r and kr f r, the remaining results follow immediately Finally, the limiting values are clear

7 QUASICONFORMAL DISTORTION FUNCTIONS 49 1 LEMMA For 1, r Ž 0, 1, let s Ž r Then the functions fž r Ž s Ž r and gž r Ž s Ž r are both strictly increasing from Ž 0, 1 onto Ž 1,, and Ž 1,1, respectiely Proof By logarithmic differentiation, fž r rr Ž r Ž r Ž s E Ž s s Ž s fž r Ž r EŽ r r Ž r, which is positive by AVV3, Theorem Ž 3, Ž 7, so that the result for f follows Next, since gž r fž r, the result for g follows immediately 13 LEMMA Suppose that fž x, y is positie, increasing, and logconcae in x and y, separately, for Ž x, y in the plane domain D, that Ž1fŽ x, yž fx is decreasing in y, and that Ž1fŽ x, yž fy is decreasing in x Then, for each segment I D parallel to the ector Ž cos, sin, Ž 0,, f I is log-concae Proof Let Ft Ž log fžxž t, yt Ž, f Ž x, y Ž1fŽ x, yž fx 1, f Ž x, y Ž1fŽ x, yž fy Here, xt tx Ž 1 t x and yt 1 ty1 Ž 1t y are simultaneously increasing or decreasing Then d F f1 f Ž x1x Ž y1y dt x y which is negative by the hypotheses / f1 f Ž x1xž y1y, ž y x For convenience, we also recall the following result Žcf AVV, Lemma 1 14 LEMMA For an interal I 0,, suppose that f, g : I 0, are functions such that fž x gž x is decreasing on I 04 and gž 0 0, gž x 0 for x 0 Then fž xy Ž gž x gž y gž xy Ž fž x fž y for x, y, x y I Moreoer, if the monotonicity of fž x gž x is strict, then the aboe inequality is also strict on I 0 4

8 50 QIU, VAMANAMURTHY, AND VUORINEN 3 PROOFS 31 Proof of Theorem 15 By logarithmic differentiation fž Ž r r s Ž s log, Ž 3 f Ž r Ž 1r where s Ž r Since s Ž s is strictly decreasing in by AVV3, Theorem Ž 3, it follows from Ž 3 that fž r r Ž r Ž r log, f Ž 1r which is negative by Lemma 1 Hence f is strictly decreasing on 1, Next, by logarithmic differentiation, we have gž Ž r r už u log, Ž 33 g Ž r 1r where u Ž r Since u Ž u is strictly increasing in by 1 AVV3, Theorem Ž 3, it follows from Ž 33 that g r log Ž r log, g 1r which is positive by the inequality LV, p 6 1r Ž r log r Hence g is strictly increasing on 1, The limiting values lim f 1 and lim g and inequalities Ž 1 and Ž are clear 34 Proof of Theorem 16 Let r th x, s r Then 1 s Ž s fž x, rž r which is strictly decreasing and positive, by Lemma 11, so that f is strictly increasing and concave The limiting values are clear Equalities in Ž 1 and Ž are all clear For the strict inequality in Ž 1 let 1 and a b, th xa, th yb, A Ž a, B Ž b, u arth Ž th x, arth Ž th y Then xy thž x y ab th 1 1 th Ž x y 1ab ab

9 QUASICONFORMAL DISTORTION FUNCTIONS 51 Similarly, u A B th 1AB AB By concavity of f, we have arth ŽthŽŽ x y Ž u, so that ŽthŽŽ x y thžž u, hence Ž 1 follows For Ž, AVV1, Lemma 315 implies that fž x x is strictly decreasing, hence fž xy fž x fž y by Lemma 14 Hence so that arth Ž thž x y arth Ž th x arth Ž th y ž / ž / Ž th x Ž th y arth, 1 Ž th x Ž th y th x th y Ž th x Ž th y 1th x th y 1 Ž th x Ž th y Now set a th x, b th y, and the first inequality in Ž follows The second inequality in Ž holds by Ž 1 35 Remark By inverting the function in Theorem 16, one sees that gž x arth Ž th x, 1, is strictly increasing and convex from Ž 0, 1 onto Ž 0,, so that the inequalities in the theorem are reversed for Proof of Theorem 17 First, writing fž x logž ss, where s x x r, r e Ž e 1, we have drdx rr and ž / 1 s ds fž x s s dx Ž s dr 1 Ž s, rr Ž r dx Ž r which is positive and strictly increasing in r from Ž 0, 1 onto Ž 1, by Lemma 1 Hence f is strictly increasing and convex, with Next, if p, q 0, 1 with p q 1, then 1 fž x Ž 37 log e pxqy f pxqy pf x qf y

10 5 QIU, VAMANAMURTHY, AND VUORINEN by the convexity of f Thus px qy x p y Ž e e Ž e Ž e If we set e x a, e y b, and p q 1, then the upper bound in Ž 1 follows If x y and 1, it follows from Ž 37 that Ž x y fž x fž y Ž xy, Ž x y by integration from y to x Now follows if we set e b and e a Next, by differentiation and simplification, we get s Ž s gž x, s Ž r which is positive and decreasing by AVV3, Theorem Ž 3, so that the assertion for g follows The lower bound in Ž 1 now follows from the convexity of g if one sets e x a and e y b Finally, the equality cases are all clear q 4 ON SOME CONJECTURES The following conjectures appear in AVV4, p 5 : Conjecture 1 th Ž r Ž th r for 1,, r 0, 1 Conjecture For each 1 define f on Ž 0, 1 by Ž r 1 fž r r Ž r Ž r For 1, f Ž r is strictly decreasing, f Ž r is the constant function, and, for, f Ž r is strictly increasing Conjecture 3 For, g Ž r Ž r 1 Ž Ž r rž r 1 1 is Ž Ž11 strictly decreasing from 0, 1 onto 1,4 Conjecture 4 For 1, r Ž 0, 1, Ž Ž r r 11 Ž 1r

11 QUASICONFORMAL DISTORTION FUNCTIONS 53 Conjecture 5 fž, r log Ž r is jointly concave as a function of two variables Ž, r on Ž 1, Ž 0, 1 In this section we obtain results which imply the truth of Conjectures 14 and give a partial solution to Conjecture 5 The following theorem generalizes the result on Conjecture 1 41 THEOREM Let a 1, be a constant, and let f : 0, a 0, 1 be an increasing differentiable function satisfying the conditions Ž i fž 0 0, Ž ii lim fž x 1, and Ž iii fž x xfž x fž x for all x Ž 0, a x a Then, for each fixed r Ž 0, 1, the function g f Ž r Ž fž r Ž is strictly decreasing from 1, onto f 1,1In particular, for all 1, and r 0, 1 f Ž 1 Ž fž r f Ž r Ž fž r Proof Let t f r, s r, u t, and h x x Ž x Ž x Then g fž s u and u sfž s g hž s hž u hž s hž u 0, fž s fž s since Ž iii implies that sfž s fž s 1, hž x is a strictly decreasing function AVV3, Theorem Ž 3, and u s by Ž iii 4 COROLLARY For r 0, 1, we hae Ž 1 g th Ž r Ž th r is strictly decreasing from 1, 1 onto ŽŽe 1 Že 1,1 In particular, e 1 e 1 th r th r th r, with equality in the first if and only if r 0 or tends to, and in the second if and only if r 0 or 1 Ž g sin Ž r Ž sin r is strictly decreasing from 1, onto Žsin 1, 1 In particular Ž sin r sin 1 sin Ž r Ž sin r, with equality in the first if and only if r 0 or tends to and in the second if and only if r 0 or 1 Ž 3 g logž1 Ž r ŽlogŽ 1 r 3 is strictly decreasing from 1, onto Žlog, 1 In particular, Ž log Ž logž 1 r log 1 Ž r Ž logž 1 r,

12 54 QIU, VAMANAMURTHY, AND VUORINEN for all 1, and r 0, 1 The first inequality reduces to equality if and only if r 0 or tends to The second inequality reduces to equality if and only if r 0 or 1 Proof Ž 1 Take fž x th x and a Then, the conditions ŽŽ i, ii, and Ž iii hold and hence the result follows Ž Take fž x sin x and a Then the conditions ŽŽ i, ii, and Ž iii also hold, and hence the result follows Ž 3 Take fž x logž 1 x and a e 1 Then again fž x satisfies all the conditions of Theorem 41, so that the result follows The next theorem indicates that Conjecture is true 43 THEOREM For each fixed 1, define function f by Ž r 1 fž r r Ž r Ž r Ž 11 Then f is strictly decreasing from 0, 1 onto,4 for Ž 1,, strictly increasing from Ž 0, 1 onto Ž4 11, if, and f Ž r Proof For Ž r, Ž 0, 1 Ž 1,, define 1 FŽ r, log fž r log s log r log Ž s log Ž r, where s Ž r Employing 8, we have F rr Ž r gž r,, Ž 44 r where g r, s E s r E r r r r r, and g Ž r GŽ r,, Ž 45 where Gr, Ž s Es s Ž s Ž rer r Ž r Next, by Lemma 10 and AVV3, Theorem Ž 3, it follows that for each r, Gr, is an increasing function of on 1, Because the function Ž rež r Ž r Ž r is strictly increasing from Ž 0, 1 onto Ž 0, Žsee AVV3, p 539, ½ 5 lim GŽ r, Ž r EŽ r r Ž r 0 Ž 46

13 QUASICONFORMAL DISTORTION FUNCTIONS 55 On the other hand, E r r r E r r r 47 for r 0, 1 since E r r r E r r r r Ž r EŽ r Ž r Ž r 0, Ž 48 by Lemma 10 Hence ½ 5 GŽ r,1 Ž r Ž r EŽ r r Ž r EŽ r r Ž r Ž r EŽ r r Ž r ½ Ž r EŽ r r Ž r 5 0 Ž 49 From Ž 46, Ž 49, the monotonicity of Gr, in, and Ž 45, it follows that, for any fixed r Ž 0, 1, there exists a unique such that 0 0, if 1, g 0, if 0, 0, if 0 Thus, for any fixed r Ž 0, 1, gž r,, as a function of, is strictly decreasing on 1, and strictly increasing on Ž, Note that Ž r 0 0 rž 1r, and gž r,1 gž r, 0 by the Landen transformation BB, p 1 Hence, 1 for any fixed r Ž 0, 1, and gž r, 0 0 if Ž 1,, gž r, 0 for, so that the result follows from Ž 44 Finally, the limiting values are clear The following result indicates that Conjecture 3 is actually contained in Conjecture 410 COROLLARY For, g Ž r Ž r 1 Ž Ž r 1 1 Ž Ž Ž1 1 r r is strictly decreasing from 0, 1 onto 1,4 For Ž 1,, g is strictly increasing from Ž 0, 1 onto Ž4 Ž11,1 Proof Set t Ž r Then g Ž r 1f Ž t 1, where f is as in Theorem 43 Hence the result follows from Theorem 43 0

14 56 QIU, VAMANAMURTHY, AND VUORINEN We now turn to the study of Conjecture 4 Ž 411 THEOREM For r 0, 1, the function H r 1 r Ž r is strictly decreasing from 1, onto Ž1, Ž 1 r In particular, for r Ž 0, 1 and 1,, Ž r Ž r 11 1 Ž 1r The equality holds if and only if 1 Proof Let t r, s Ž r, u Ž t Then, by 9, 8 sž s H H 1Ž r,, Ž 41 H where u u 1 log 1 r H1Ž r, Ž t Ž r sž s 4 sž s 4 Let H r, u u s s, h x x x E x Then 1 H hž s hž u Ž HŽ r, The function hž x is strictly increasing from Ž 0, 1 onto Ž 0,, by AVV3, Theorem ŽŽ 3, 7 Hence it follows from Ž 413 that H 0 for r Ž 0, 1 and Ž 1,, since s u for r Ž 0, 1 and Ž 1, So H Ž r,, as a function of, is strictly decreasing on 1,, and so is H Ž r, 1

15 QUASICONFORMAL DISTORTION FUNCTIONS 57 Now it follows from 41 that 8 sž s HŽ H H r,1 1 ½ 1 1 t Ž t Ž t r Ž r Ž r 4 rž r ½ t t t log t 4 rž r logž 1 r5 1 r Ž r Ž r log r logž 1 r 5 ½ 5 r Ž r Ž r log r logž 1 r, 8 rž r which is negative for r Ž 0, 1 by AVV3, Lemma 4Ž 1 and Lemma 1, since t r Hence H is strictly decreasing in on 1, The limiting values and the remaining conclusions in the theorem are clear By the same method used in the proof of Theorem 411, we can obtain the following improvement of the inequality stated in Conjecture 4 Ž r 414 THEOREM For each r 0, 1, g r 1 1r Ž r is strictly decreasing from 1, onto Ž1, Ž 1 1r In particular, for any r Ž 0, 1 and 1,, Ž r 11 Ž r Ž 1 1r, with equality if and only if 1 Proof Let t r, s Ž r, u Ž t Then, by differentiation, g r G r Ž, 8 sž s g r

16 58 QIU, VAMANAMURTHY, AND VUORINEN where u Ž u 1 logž 1 t Gr Ž t Ž r s Ž s 4 sž s By a method similar to the proof of Theorem 411, we can get 1 Gr ½ t Ž t Ž t r Ž r Ž r 4 rž r logž 1 t5 ½ 5 t t t log t log 1 t 0, 8 rž r yielding the monotonicity of g Ž r The remaining conclusions are clear Finally, we prove a result about the Conjecture THEOREM Define the function f on Ž 1, Ž 0, 1 by fž r, log Ž r Then, for each segment I D parallel to the ector Ž cos, sin with Ž 0,,f I is concae In particular, if t, r, r Ž 0, 1 1, 1, 1, with r r and or with r r and, then t 1t Ž 1 t Ž1t r r tr 1 t r Proof Clearly, Ž r is increasing in r as well as in By Q, Lemma 1; AVV4, Corollary Ž 1 and Ž, Ž r is log-concave in r and in, separately We can see from 8 and 9 that Ž 1sŽ s r is decreasing in, and Ž 1sŽ s is decreasing in r, where s Ž r Hence the result follows from Lemma THEOREM For r Ž 0, 1, c Ž 1,, let f be defined on 1, by Ž 1c f u u s s, where u t, s r, t r Then f is decreasing, with lim f 0 Proof By differentiation and simplification, we see that fž 0on 1, if and only if hs hu, where h x x E x x Ž x But this is true since h is strictly increasing AVV3, Theorem Ž 3 and since s u

17 QUASICONFORMAL DISTORTION FUNCTIONS 59 5 PROPERTIES OF THE -DISTORTION FUNCTION As is well known, the special function ž / Ž introduced by Lehto, Virtanen, and Vaisala LVV also plays an important role in the distortion theory of plane quasiconformal mappings It yields, for example, a sharp upper bound for the linear dilatation of a plane quasiconformal mapping, and measures the distortion of the boundary values of a -quasiconformal self-mapping of the upper half plane preserving the point In this section, we study some properties of Ž 5 THEOREM The function fž log Ž log is decreasing on Ž 0, 1 and increasing on 1,, onto a,, where a In particular, for Ž 0,, Ž a Proof Since fž 1 fž, it is enough to prove the theorem for 1 Ž 1, For this, set r Ž 1 ŽŽ Then 1 r 1 and Ž log r log r f log Ž r log Ž r Let hr logž rr, pr logž Ž r Ž r, gž r hrpr Then hž 1 pž 1 0, and hž r 4 Ž r Ž r, pž r which is strictly increasing in r on 1, 1 by AVV3, Theorem Ž 8 Hence g is also increasing there by the Monotone l Hopital s ˆ Rule AVV5, Lemma, and so is f The limiting values are clear 53 COROLLARY For Ž 0,, define F on Ž 0, by FŽ p Ž p 1 p Then Ž a 1 For 1,, F p is increasing from 0, onto,, where Ž a 4 1 In particular, for Ž 1,, a p 1p Ž if p Ž 0,1, Ž 54

18 60 QIU, VAMANAMURTHY, AND VUORINEN and a p 1p Ž if p Ž 1, Ž 55 Ž For Ž 0, 1, FŽ p is decreasing from Ž 0, onto Ž0, a, where aisasinž 1In particular, for Ž 0, 1, the inequalities Ž 54 and Ž 55 are both reersed Proof For Ž 1, since log Ž p log FŽ p log, p log the results follow from Theorem 5 For Ž, set L 1 Then FŽ p 1ŽL p 1p, and the results follow by Ž 1 56 THEOREM For p Ž 0,, the function GŽ Ž p Ž p is strictly decreasing on Ž 0, if p Ž 1,, and strictly increasing on Ž 0, if pž 0, 1, and it has range Ž 0, In particular, p Ž p, if 1orp1, p Ž, if, p Ž 1, or, p Ž 0,1 p, if 1, p 1, or 1, p 0,1, Proof That the range is Ž 0,, follows from AVV1, Theorem 13 1 Next, set r ŽŽ p, rr Then, by Ž 51, Ž 1, and 7, p 1 Ž p r 1 r, Ž r Ž p, Ž 57 p p p drp p rpž 1 rp Ž rp Ž r p Ž 58 d Clearly, 1 r, if, p Ž 1,, r, if p1, r 1 p Ž 59 r, if 1, 1 and r, if 1, p Ž 1,, rp 1, if 1, p Ž 0,1, r Ž 510 r 1, if,pž 0,1 p

19 QUASICONFORMAL DISTORTION FUNCTIONS 61 By 57, 58, and differentiation, we get G Ž r Ž r Ž r p Ž r p 4p G The results now follow from this equality, Ž 59, Ž 510, and AVV3, Theorem Ž THEOREM For each Ž 1, let f be defined on Ž 0, by fž t Ž t Ž 1tThen f is strictly decreasing on Ž0, 1 and strictly increasing on 1, In particular, for t Ž 0, and 1, t 1t, with equality if and only if t 1 or 1 Proof Denote r tž t 1, s Ž r, u Ž r, and let gž r Ž 1 log fž t log s log s log u log u Note that t 1 if and only if r 1 Then, by differentiation, ž / ž / 1 s ds 1 u du gž r s s dr u u dr 5 1 ½ Ž s Ž u, rr Ž r Ž r which is negative for r r and positive for r r, by Lemma 1 51 Remark It is easy to show that, for each Ž 1,, the function fž p Ž p Ž p Ž p, is increasing from Ž 0, onto Ž a,, where Ž a 4 1 In particular, a fž 1 fž p, for all, p Ž 1, 6 APPLICATIONS We can make use of the above theorems to improve some existing results in the distortion theory of quasiconformal mappings In this section, we only show some examples 61 THEOREM Suppose that f is a -quasiconformal mapping of the unit Ž disk B onto itself with f 0 0 Then Ž 1 Ž 1 1z z fž z Ž1 1z z 1, for all z B; Ž Žsinz sin fž z Žsinz, for all z B; 1 1 Ž11

20 6 QIU, VAMANAMURTHY, AND VUORINEN Ž 3 Žthz th fž z Žthz 1, for all z B; Ž 4 ŽlogŽ1 z logž1 fž z ŽlogŽ1 z 1, for all z B; Ž w1w 1x1 1x 1 1Ž x 1 z Ž 1z for z 1, z B, where wj f z j, xj 1x j 1, j1,, x Žz z, and x z z ŽŽ z z Proof Part Ž 1 follows from Theorem 15 and the well-known quasiconformal Schwarz lemma Ž z fž z Ž z 1 Parts ŽŽ, 3, and Ž 4 follow from Corollary 4, so long as we note that all the right inequalities in Corollary 4 are reversed if is replaced by 1 For Ž 5, we employ the following inequality AVV4, Lemma Ž x 5 Ž 1Ž 11 w w w w Ž x, 1 1 ½ from which we obtain, by Theorem 15, Ž w w z z 1x 1 1 Ž x ½ 5 1x Ž 1Ž 11 1 x 1 Ž 1 1 x 1 11 Ž x 1 1 z z 1 1 x 1 Ž 1 1x 1x 1 Ž 1 1 x 1 11 z z 1 1

21 QUASICONFORMAL DISTORTION FUNCTIONS 63 6 Remark Theorem 15 can also be used to obtain explicit bounds Ž for sin f z,th f z, and log 1 f z in Corollary 61, 3, and 4 ACNOWLEDGMENT The authors are indebted to Professor G D Anderson for his valuable remarks on this paper REFERENCES A S Agard, Distortion theorems for quasiconformal mappings, Ann Acad Sci Fenn Ser A I Math 413 Ž 1968, 11 AVV1 G D Anderson, M Vamanamurthy, and M Vuorinen, Distortion functions for plane quasiconformal mappings, Israel J Math 6 Ž 1988, 116 AVV G D Anderson, M Vamanamurthy, and M Vuorinen, Special functions of quasiconformal theory, Exposition Math 7 Ž 1989, AVV3 G D Anderson, M Vamanamurthy, and M Vuorinen, Functional inequalities for complete elliptic integrals and their ratios, SIAM J Math Anal 1 Ž 1990, AVV4 G D Anderson, M Vamanamurthy, and M Vuorinen, Inequalities for plane quasiconformal mappings, in The Mathematical Heritage of Wilhelm MagnusGroups, Geometry and Special Functions ŽW Abikoff, J S Birman, and uiken, Eds, pp 17, Contemp Math, Vol 169, Amer Math Soc, Providence, RI, 1994 AVV5 G D Anderson, M Vamanamurthy, and M Vuorinen, Inequalities for quasiconformal mappings in space, Pacific J Math 160 Ž 1993, 118 AVV6 G D Anderson, M Vamanamurthy, and M Vuorinen, Functional inequalities for hypergeometric functions and complete elliptic integrals, SIAM J Math Anal 3 Ž 199, 5154 B B C Berndt, Ramanujan Notebooks, Part III, Springer-Verlag, BerlinHeidelbergNew York, 1991 BB J M Borwein and P B Borwein, Pi and the AGM, Wiley, New York, 1987 BF P F Byrd and M D Friedman, Handbook of elliptic integrals for engineers and physicists, Grundlehren Math Wiss, Vol 57, Springer-Verlag, Berlin GottingenHeidelberg, 1954 He C-Q He, Distortion estimates of quasiconformal mappings, Sci Sinica Ser A 7 Ž 1984, 53 LV O Lehto and I Virtanen, Quasiconformal Mappings in the Plane, nd ed, Grundlehren Math Wiss, Vol 16, Springer-Verlag, New York HeidelbergBerlin, 1973 LVV O Lehto, I Virtanen, and J Vaisala, Contributions to the distortion theory of quasiconformal mappings, Ann Acad Sci Fenn Ser A I Math 73 Ž 1959, 114 P D Partyka, Approximation of the Hersch-Pfluger distortion function-applications, Ann Uni Mariae Curie-Sklodowska 45, No 1 Ž 199, Q S-L Qiu, Distortion properties of -qc maps and a better estimate of Mori s constant, Acta Math Sinica 35 Ž 199, 49504

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