# Monotonicity, Convexity, and Inequalities Involving the Agard Distortion Function

Authors: Yu-Ming Chu; Miao-Kun Wang; Yue-Ping Jiang; Song-Liang Qiu
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We present some monotonicity, convexity, and inequalities for the Agard distortion functionηK(t) and improve some well-known results.


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## Body

## 1. Introduction

Forr∈[0,1], Lengedre’s complete elliptic integrals of the first and second kind [1] are defined by (1.1)K=K(r)=∫0π/2(1-r2sin⁡2θ)-1/2dθ,K′(r)=K(r′),K(0)=π2,K(1)=∞,(1.2)E=E(r)=∫0π/2(1-r2sin⁡2θ)1/2dθ,E′(r)=E(r′),E(0)=π2,E(1)=1,
respectively. Here and in what follows, we set r′=1-r2.Letμ(r) be the modulus of the plan Grötzsch ring B2∖[0,r] for r∈(0,1), where B2 is the unit disk. Then, it follows from [2] that
(1.3)μ(r)=π2K′(r)K(r).ForK∈(0,∞), the Hersch-Pfluger distortion function φK(r) is defined as
(1.4)φK(r)=μ-1(μ(r)K)forr∈(0,1),φK(0)=φK(1)-1=0,
while the Agard distortion function ηK(t) and the linear distortion function λ(K) are defined by
(1.5)ηK(t)=[φK(r)φ1/K(r′)],λ(K)=ηK(1),r=t1+tt∈(0,∞),
respectively.It is well known that the functionsηK(t) and λ(K) play a very important role in quasiconformal theory, quasiregular theory, and some other related fields [3–8]. For example, Martin [8] found that the sharp upper bound in Schottky’s theorem can be expressed by ηK(t), and in [9–15] the authors established a number of remarkable properties for the Agard distortion function ηK(t).In [14], the authors proved that
(1.6)eπ(K-1)<λ(K)<ea(K-1),(1.7)eb(K-1/K)<λ(K)<eπ(K-1/K)
for all K∈(1,∞), where a=(4/π)𝒦(1/2)2=4.3768…, b=a/2. Recently, Anderson et al. [15] established that
(1.8)λ(K)<e(π+b/K)(K-1),(1.9)e[log⁡2+(a-log⁡2)/K](K-1)<λ(K)<e[π+(a-log⁡2)/K](K-1)
for all K∈(1,∞), where a and b are defined as in inequalities (1.6) and (1.7), respectively.The purpose of this paper is to present the new monotonicity, convexity, and inequalities for the Agard distortion functionηK(t) and improve inequalities (1.6)–(1.9).Our main results are Theorems1.1 and 1.2 as follows.Theorem 1.1.
LetK∈(1,∞), a=(4/π)𝒦(1/2)2=4.3768…, b=a/2, and c∈ℝ. Then, the following statements are true.(1)
f(K)=λ(K)/Kc is strictly increasing from (1,∞) onto (1,∞) for c≤a; if c>a, then there exists K0∈(1,∞), such that f is strictly decreasing in (1,K0) and strictly increasing in (K0,∞). In particular, the inequality λ(K)≥Kc holds for all K∈(1,∞) with the best possible constant c=a.(2)
g(K)=[logηK(t)-logt]/(K-1) is convex in (1,∞) for fixed t∈(0,∞).(3)
Ift≥1 and r=t/(1+t), then h(K)=[logηK(t)-logt]/(K-1/K) is strictly increasing from (1,∞) onto (2𝒦(r)𝒦′(r)/π,π𝒦(r)/𝒦′(r)).Theorem 1.2.
Lett∈(0,∞), r=t/(1+t), a=(4/π)𝒦(1/2)2,b=a/2, A(r)=π2/(2μ(r)), B(r)=8𝒦(r)𝒦′(r)2[ℰ(r)-r′2𝒦(r)]/π2, and Fc(K)=K[(logηK(t)-logt)/(K-1)-c]. Then, the following statements are true.(1)
Fc(K) is strictly decreasing from (1,∞) onto (-∞,4𝒦(r)𝒦′(r)/π-c) for c>A(r). If c=A(r), then Fc(K) is strictly decreasing from (1,∞) onto (A(r)-4log2-logt,4𝒦(r)𝒦′(r)/π-A(r)). Moreover,
(1.10)te(K-1)(A(r)+((A(r)-4log⁡2-log⁡t)/K))<ηK(t)<te(K-1)(A(r)+((4K(r)K′(r)/π-A(r))/K))
for all t∈(0,∞) and K∈(1,∞). In particular, if t=1, then (1.10) becomes
(1.11)e(K-1)(π+((π-4log⁡2)/K))<λ(K)<e(K-1)(π+((a-π)/K)).(2)
Ifc≤B(r), then Fc(K) is strictly increasing from (1,∞) onto (4𝒦(r)𝒦′(r)/π-c,∞). Moreover,
(1.12)ηK(t)>te(K-1)(B(r)+((4K(r)K′(r)/π-B(r))/K))
for all t∈(0,∞) and K∈(1,∞). In particular, if t=1, then (1.12) becomes
(1.13)λ(K)>e(K-1)(b+(b/K))=eb(K-(1/K)).(3)
IfB(r)<c<A(r), then there exists K1∈(1,∞) such that Fc(K) is strictly decreasing on (1,K1) and strictly increasing on (K1,∞).(4)
Fc(K) is convex in (1,∞).


## 2. Lemmas

In order to prove our main results, we need several formulas and lemmas, which we present in this section.The following formulas were presented in [14, Appendix  E, pp. 474-475]. Let t∈(0,∞),K∈(0,∞), r=t/(1+t)∈(0,1), and s=φK(r). Then,
(2.1)dK(r)dr=E(r)-r′2K(r)rr′2,dE(r)dr=E(r)-K(r)r,K(r)E′(r)+K′(r)E(r)-K(r)K′(r)=π2,dμ(r)dr=-π24rr′2K(r)2,∂s∂r=ss′2K(s)K′(s)rr′2K(r)K′(r),∂s∂K=2πKss′2K(s)K′(s),φK(r)2+φ1/K(r′)2=1,ηK(t)=(ss′)2,∂ηK(t)∂K=4πKηK(t)K(s)K′(s)=2μ(r)K′(s)2ηK(t).Lemma 2.1 (see [14, Theorem  1.25]).
For-∞<a<b<∞, let f,g:[a,b]→ℝ be continuous on [a,b] and differentiable on (a,b), and let g′(x)≠0 on (a,b). If f′(x)/g′(x) is increasing (decreasing) on (a,b), then so are
(2.2)f(x)-f(a)g(x)-g(a),f(x)-f(b)g(x)-g(b).
If f′(x)/g′(x) is strictly monotone, then the monotonicity in the conclusion is also strict.The following lemma can be found in [14, Theorem  3.21(1) and (7), Lemma  3.32(1) and Theorem  5.13(2)].Lemma 2.2.
(1)[ℰ(r)-r′2𝒦(r)]/r2 is strictly increasing from (0,1) onto (π/4,1);
(2)r′c𝒦(r) is strictly decreasing from (0,1) onto (0,π/2) if and only if c≥1/2;
(3)𝒦(r)𝒦′(r) is strictly decreasing in (0,2/2) and strictly increasing in (2/2,1);
(4)μ(r)+logr is strictly decreasing from (0,1) onto (0,log4).Lemma 2.3.
Letr∈[1/2,1), K∈(1,∞), and s=φK(r). Then, G(K)≡{π/[2𝒦(s)]}2+[μ(r)/𝒦′(s)]2 is strictly decreasing from (1,∞) onto (𝒦′(r)2/𝒦(r)2,π2/[2𝒦(r)2]).Proof.
ClearlyG(1+)=π2/(2𝒦(r)2), G(+∞)=𝒦′(r)2/𝒦(r)2. Differentiating G(K), one has
(2.3)G′(K)=4πKμ(r)2K(s)K′(s)-2[E′(s)-s2K′(s)]-πKK(s)-2K′(s)[E(s)-s′2K(s)]=4πKK(s)-2K′(s)-2G1(K),
where G1(K)=[ℰ′(s)-s2𝒦′(s)]𝒦(s)3μ(r)2-π2[ℰ(s)-s′2𝒦(s)]𝒦′(s)3/4.
From Lemma2.2(1) and (2), we clearly see that G1(K) is strictly decreasing in (1,∞). Moreover,
(2.4)lim⁡K→1+G1(K)=[E′(r)-r2K′(r)]K(r)3μ(r)2-π24[E(r)-r′2K(r)]K′(r)3=π24K′(r)2G2(r),
where G2(r)=𝒦(r)[ℰ′(r)-r2𝒦′(r)]-𝒦′(r)[ℰ(r)-r′2𝒦(r)] is also strictly decreasing in (0,1). Thus, G2(r)≤G2(2/2)=0 for r∈[1/2,1), and G1(K)<G1(1+)≤0 for K∈(1,∞).
Therefore, the monotonicity ofG(K) follows from (2.3) and (2.4) together with the fact that G1(K)<0 for K∈(1,∞).


## 3. Proofs of Theorems1.1 and 1.2

Proof of Theorem1.1.
For part (1), clearlyf(1+)=1. Let r=μ-1[π/(2K)] for K∈(1,∞), then λ(K)=(r/r′)2, r∈(1/2,1),
(3.1)drdK=2πrr′2K′(r)2,dλ(K)dK=4πλ(K)K′(r)2,(3.2)lim⁡K→+∞f(K)=lim⁡r→1r2K′(r)r′2K(r)=+∞.
Making use of (3.1), we have
(3.3)Kc+1f′(K)λ(K)=f1(K)≡4πK′(r)K(r)-c.
It follows from Lemma2.2(3) that f1(K) is strictly increasing from (1,∞) onto (a-c,∞). Then, from (3.2) and (3.3), we know that f is strictly increasing from (1,∞) onto (1,∞) for c≤a. If c>a, then there exists K0∈(1,∞) such that f is strictly decreasing in (1,K0) and strictly increasing in (K0,∞). Moreover, the inequality λ(K)≥Kc holds for all K∈(1,∞) with the best possible constant c=a.
For part (2), denoter=t/(1+t). Differentiating g(K), we get
(3.4)g′(K)=2K′(s)2(K-1)/μ(r)-(log⁡ηK(t)-log⁡t)(K-1)2.
Letg1(K)=2𝒦′(s)2(K-1)/μ(r)-(logηK(t)-logt) and g2(K)=(K-1)2, then g1(1)=g2(1)=0, g′(K)=g1(K)/g2(K) and
(3.5)g1′(K)g2′(K)=g3(K)≡-2μ(r)2[E′(s)-s2K′(s)]K′(s)3.
Clearly, g3(K) is strictly increasing in (1,∞). Then, (3.5) and Lemma 2.1 lead to the conclusion that g′(K) is strictly increasing in (1,∞). Therefore, g(K) is convex in (1,∞).
For part (3), ift≥1, then r≥2/2. Let h1(K)=logηK(t)-logt and h2(K)=K-1/K, then h1(1)=h2(1)=0, h(K)=h1(K)/h2(K), and
(3.6)h1′(K)h2′(K)=2K′(s)2/μ(r)1+K-2=2μ(r)G(K),
where G(K) is defined as in Lemma 2.2.
Therefore,h(K) is strictly increasing in (1,∞) for t≥1 follows from Lemmas 2.1 and 2.2 together with (3.6). Moreover, making use of l’Hôpital’s rule, we have h(1+)=2𝒦(r)𝒦′(r)/π, h(∞)=π𝒦(r)/𝒦′(r).Proof of Theorem1.2.
DifferentiatingFc(K) gives
(3.7)Fc′(K)=log⁡ηK(t)-log⁡tK-1-c+K[(2K′(s)2(K-1))/μ(r)-(log⁡ηK(t)-log⁡t)(K-1)2]=2K′(s)2K(K-1)/μ(r)-(log⁡ηK(t)-log⁡t)(K-1)2-c.
Let
(3.8)H(K)=[2K′(s)2K(K-1)/μ(r)]-[log⁡ηK(t)-log⁡t](K-1)2,H1(K)=2𝒦′(s)2K(K-1)/μ(r)-(logηK(t)-logt), and H2(K)=(K-1)2, then H(K)=H1(K)/H2(K), H1(1)=H2(1)=0, and
(3.9)H1′(K)H2′(K)=H3(K)≡4πμ(r)[E(s)-s′2K(s)]K′(s)3.
Clearly, thatH3(K) is strictly increasing in (1,∞) follows from Lemma 2.2(1) and (2). Then, from (3.8) and (3.9) together with Lemma 2.1, we know that H(K) is strictly increasing in (1,∞). Moreover, l’Hôpital’s rule leads to
(3.10)lim⁡K→1H(K)=B(r),lim⁡K→∞H(K)=A(r).
For part (1), ifc>A(r), then from (3.7) and (3.8), we know that Fc′(K)<0 for K∈(1,∞) and Fc(K) is strictly decreasing in (1,∞). Moreover,
(3.11)lim⁡K→1Fc(K)=[4K(r)K′(r)/π]-c,lim⁡K→∞Fc(K)=-∞.
If c=A(r), then Fc(K) is also strictly decreasing in (1,∞) and Fc(1+)=[4𝒦(r)𝒦′(r)/π]-A(r), and from Lemma 2.2(4) we get
(3.12)lim⁡K→∞Fc(K)=lim⁡K→∞KK-1[-2log⁡(s′)-2μ(s′)+A(r)-log⁡t]=A(r)-4log⁡2-log⁡t.
Therefore, inequalities (1.10) and (1.11) follows from (3.12) and the monotonicity of Fc(K) when c=A(r).
For part (2), ifc≤B(r), then that Fc(K) is strictly increasing in (1,∞) follows from (3.7) and (3.8). Note that
(3.13)lim⁡K→1Fc(K)=[4K(r)K′(r)/π]-c,lim⁡K→∞Fc(K)=+∞.
Therefore, inequalities (1.12) and (1.13) follow from (3.13) and the monotonicity of Fc(K) when c=B(r).
For part (3), ifB(r)<c<A(r), then from (3.7) and (3.8) together with the monotonicity of H(K) we clearly see that there exists K1∈(1,∞), such that Fc′(K)<0 for K∈(1,K1) and Fc′(K)>0 for K∈(K1,∞). Hence, Fc(K) is strictly decreasing in (1,K1) and strictly increasing in (K1,∞).
Part (4) follows from (3.7) and (3.8) together with the monotonicity of H(K).Takingt=1 in Theorem 1.2, we get the following corollary.Corollary 3.1.
Leta and b be defined as in Theorem 1.2, c∈ℝ, and fc(K)=K{[logλ(K)]/(K-1)-c}. Then,(1)
ifc>π, then fc(K) is strictly decreasing from (1,∞) onto (-∞,a-c); if c=π, then fc(K) is strictly decreasing from (1,∞) onto (π-4log2,a-π);(2)
ifc≤b, then fc(K) is strictly increasing from (1,∞) onto (a-c,∞);(3)
ifb<c<π, then there exists K2∈(1,∞), such that fc(K) is strictly decreasing in (1,K2) and strictly increasing in (K2,∞);(4)
fc(K) is convex in (1,∞).Inequalities (1.11) and (1.13) lead to the following corollary, which improve inequalities (1.6)–(1.9).Corollary 3.2.
Leta and b be defined as in Theorem 1.2, then the following inequality
(3.14)max⁡{e(K-1)(π+((π-4log⁡2)/K)),eb(K-(1/K))}<λ(K)<e(K-1)(π+((a-π)/K)).
holds for all K∈(1,∞).



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*Source: 671765-2011-12-15.xml*