Monotonicity, Convexity, and Inequalities Involving the Agard Distortion Function

Abstract

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

1. Introduction

For 𝑟∈[0,1], Lengedre’s complete elliptic integrals of the first and second kind [1] are defined by 𝒦𝒦=𝒦(𝑟)=^0𝜋/21−𝑟²sin²𝜃^−1/2𝑑𝜃,_(𝑟)=𝒦𝑟^,𝒦(0)=𝜋ℰ=ℰ(𝑟)=_0𝜋/22,𝒦(1)=∞,(1.1)ℰ1−𝑟²sin²𝜃^1/2𝑑𝜃,_(𝑟)=ℰ𝑟^,ℰ(0)=𝜋2,ℰ(1)=1,(1.2) respectively. Here and in what follows, we set 𝑟^=√1−𝑟².

Let 𝜇(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 𝜇(𝑟)=𝜋2𝒦^(𝑟)𝒦(𝑟).(1.3)

For 𝐾∈(0,∞), the Hersch-Pfluger distortion function 𝜑_𝐾(𝑟) is defined as 𝜑_𝐾(𝑟)=𝜇⁻¹𝜇𝐾(𝑟)for𝑟∈(0,1),𝜑^𝐾(0)=𝜑_𝐾(1)−1=0,(1.4) while the Agard distortion function 𝜂_𝐾(𝑡) and the linear distortion function 𝜆(𝐾) are defined by 𝜂_𝐾(𝑡)=𝜑^𝐾(𝑟)𝜑_1/𝐾(𝑟^),𝜆(𝐾)=𝜂^𝐾(1),𝑟=𝑡1+𝑡𝑡∈(0,∞),(1.5) respectively.

It is well known that the functions 𝜂_𝐾(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 𝜂_𝐾(𝑡), and in [9–15] the authors established a number of remarkable properties for the Agard distortion function 𝜂_𝐾(𝑡).

In [14], the authors proved that 𝑒𝑒^{𝜋(𝐾−1)}<𝜆(𝐾)<𝑒^{𝑎(𝐾−1)},(1.6)_{𝑏(𝐾−1/𝐾)}<𝜆(𝐾)<𝑒^{𝜋(𝐾−1/𝐾)}(1.7) for all 𝐾∈(1,∞), where 𝑎=(4/𝜋)𝒦(1/√2)²=4.3768…, 𝑏=𝑎/2. Recently, Anderson et al. [15] established that 𝑒𝜆(𝐾)<𝑒^{(𝜋+𝑏/𝐾)(𝐾−1)},(1.8)_{[log2+(𝑎−log2)/𝐾](𝐾−1)}<𝜆(𝐾)<𝑒^{[𝜋+(𝑎−log2)/𝐾](𝐾−1)}(1.9) 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 function 𝜂_𝐾(𝑡) and improve inequalities (1.6)–(1.9).

Our main results are Theorems 1.1 and 1.2 as follows.

Theorem 1.1. Let 𝐾∈(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)/𝐾^𝑐 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)≥𝐾^𝑐 holds for all 𝐾∈(1,∞) with the best possible constant 𝑐=𝑎.(2)𝑔(𝐾)=[log𝜂_𝐾(𝑡)−log𝑡]/(𝐾−1) is convex in (1,∞) for fixed 𝑡∈(0,∞).(3)If 𝑡≥1 and 𝑟=√𝑡/(1+𝑡), then ℎ(𝐾)=[log𝜂_𝐾(𝑡)−log𝑡]/(𝐾−1/𝐾) is strictly increasing from (1,∞) onto (2𝒦(𝑟)𝒦^(𝑟)/𝜋,𝜋𝒦(𝑟)/𝒦^(𝑟)).

Theorem 1.2. Let 𝑡∈(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 𝐹_𝑐(𝐾)=𝐾[(log𝜂_𝐾(𝑡)−log𝑡)/(𝐾−1)−𝑐]. Then, the following statements are true.(1)𝐹_𝑐(𝐾) is strictly decreasing from (1,∞) onto (−∞,4𝒦(𝑟)𝒦′(𝑟)/𝜋−𝑐) for 𝑐>𝐴(𝑟). If 𝑐=𝐴(𝑟), then 𝐹_𝑐(𝐾) is strictly decreasing from (1,∞) onto (𝐴(𝑟)−4log2−log𝑡,4𝒦(𝑟)𝒦^(𝑟)/𝜋−𝐴(𝑟)). Moreover, 𝑡𝑒^{(𝐾−1)(𝐴(𝑟)+((𝐴(𝑟)−4log2−log𝑡)/𝐾))}<𝜂_𝐾(𝑡)<𝑡𝑒^{(𝐾−1)(𝐴(𝑟)+((4𝒦(𝑟)𝒦′(𝑟)/𝜋−𝐴(𝑟))/𝐾))}(1.10) for all 𝑡∈(0,∞) and 𝐾∈(1,∞). In particular, if 𝑡=1, then (1.10) becomes 𝑒^{(𝐾−1)(𝜋+((𝜋−4log2)/𝐾))}<𝜆(𝐾)<𝑒^{(𝐾−1)(𝜋+((𝑎−𝜋)/𝐾))}.(1.11)(2)If 𝑐≤𝐵(𝑟), then 𝐹_𝑐(𝐾) is strictly increasing from (1,∞) onto (4𝒦(𝑟)𝒦^(𝑟)/𝜋−𝑐,∞). Moreover, 𝜂_𝐾(𝑡)>𝑡𝑒^{(𝐾−1)(𝐵(𝑟)+((4𝒦(𝑟)𝒦′(𝑟)/𝜋−𝐵(𝑟))/𝐾))}(1.12) for all 𝑡∈(0,∞) and 𝐾∈(1,∞). In particular, if 𝑡=1, then (1.12) becomes 𝜆(𝐾)>𝑒^{(𝐾−1)(𝑏+(𝑏/𝐾))}=𝑒^{𝑏(𝐾−(1/𝐾))}.(1.13)(3)If 𝐵(𝑟)<𝑐<𝐴(𝑟), then there exists 𝐾₁∈(1,∞) such that 𝐹_𝑐(𝐾) is strictly decreasing on (1,𝐾₁) and strictly increasing on (𝐾₁,∞).(4)𝐹_𝑐(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=𝜑_𝐾(𝑟). Then, 𝑑𝒦(𝑟)𝑑𝑟=ℰ(𝑟)−𝑟^2𝒦(𝑟)𝑟𝑟^2,𝑑ℰ(𝑟)𝒦(𝑟)ℰ_𝑑𝑟=ℰ(𝑟)−𝑟𝒦(𝑟),(𝑟)+𝒦^(𝑟)ℰ(𝑟)−𝒦(𝑟)𝒦^(𝑟)=𝜋𝑑𝜇(𝑟)2,𝑑𝑟=−4𝑟𝑟^2𝜋𝒦²(𝑟)²𝜕𝑠,𝜕𝑟=𝑠𝑠^2𝒦(𝑠)𝒦^(𝑠)𝑟𝑟^2𝒦(𝑟)𝒦^(𝑟),𝜕𝜕𝐾𝑠=𝜋2𝐾𝑠𝑠^2𝜑𝒦(𝑠)𝒦^(𝑠),_𝐾𝜂(𝑟)²+𝜑_1/𝐾𝑟^²=1,_𝐾(𝑡)=𝑠𝑠^²,𝜕𝜂^𝐾(𝑡)𝜕𝐾=𝜋4𝐾𝜂^𝐾(𝑡)𝒦(𝑠)𝒦^(𝑠)=2𝜇(𝑟)𝒦^(𝑠)²𝜂_𝐾(𝑡).(2.1)

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 𝑓(𝑥)−𝑓(𝑎)𝑔(𝑥)−𝑔(𝑎),𝑓𝑔((𝑥𝑥))−−𝑓𝑔((𝑏𝑏)).(2.2) 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. Let 𝑟∈[1/√2,1), 𝐾∈(1,∞), and 𝑠=𝜑_𝐾(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. Clearly 𝐺(1⁺)=𝜋²/(2𝒦(𝑟)²), 𝐺(+∞)=𝒦^(𝑟)²/𝒦(𝑟)². Differentiating 𝐺(𝐾), one has 𝐺^(𝐾)=4𝜋𝐾𝜇(𝑟)²−𝜋𝒦(𝑠)𝒦^(𝑠)⁻²ℰ^(𝑠)−𝑠²𝒦^(𝑠)𝐾𝒦(𝑠)⁻²=4𝒦^(𝑠)ℰ(𝑠)−𝑠^2𝒦(𝑠)𝜋𝐾𝒦(𝑠)⁻²𝒦^(𝑠)⁻²𝐺₁(𝐾),(2.3) where 𝐺₁(𝐾)=[ℰ^(𝑠)−𝑠²𝒦^(𝑠)]𝒦(𝑠)³𝜇(𝑟)²−𝜋²[ℰ(𝑠)−𝑠^2𝒦(𝑠)]𝒦^(𝑠)³/4.
From Lemma 2.2(1) and (2), we clearly see that 𝐺₁(𝐾) is strictly decreasing in (1,∞). Moreover, 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),(2.4) 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 of 𝐺(𝐾) follows from (2.3) and (2.4) together with the fact that 𝐺₁(𝐾)<0 for 𝐾∈(1,∞).

3. Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1. For part (1), clearly 𝑓(1⁺)=1. Let 𝑟=𝜇⁻¹[𝜋/(2𝐾)] for 𝐾∈(1,∞), then 𝜆(𝐾)=(𝑟/𝑟^)², 𝑟∈(1/√2,1), 𝑑𝑟𝑑𝐾=𝜋2𝑟𝑟^2𝒦^(𝑟)²,𝑑𝜆(𝐾)𝑑𝐾=𝜋4𝜆(𝐾)𝒦^lim_𝐾→+∞(𝑟)²,(3.1)𝑓(𝐾)=lim_𝑟→1𝑟²𝒦^(𝑟)𝑟^2𝒦(𝑟)=+∞.(3.2) Making use of (3.1), we have 𝐾^𝑐+1𝑓^(𝐾)𝜆(𝐾)=𝑓¹(𝐾)≡4𝜋𝒦^(𝑟)𝒦(𝑟)−𝑐.(3.3)
It follows from Lemma 2.2(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)≥𝐾^𝑐 holds for all 𝐾∈(1,∞) with the best possible constant 𝑐=𝑎.
For part (2), denote 𝑟=√𝑡/(1+𝑡). Differentiating 𝑔(𝐾), we get 𝑔^(𝐾)=2𝒦^(𝑠)²(𝐾−1)/𝜇(𝑟)−log𝜂_𝐾(𝑡)−log𝑡(𝐾−1)².(3.4)
Let 𝑔₁(𝐾)=2𝒦^(𝑠)²(𝐾−1)/𝜇(𝑟)−(log𝜂_𝐾(𝑡)−log𝑡) and 𝑔₂(𝐾)=(𝐾−1)², then 𝑔₁(1)=𝑔₂(1)=0, 𝑔′(𝐾)=𝑔₁(𝐾)/𝑔₂(𝐾) and 𝑔_1(𝐾)𝑔_2(𝐾)=𝑔³(𝐾)≡−2𝜇(𝑟)²ℰ^(𝑠)−𝑠²𝒦^(𝑠)𝒦^(𝑠)³.(3.5) 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), if 𝑡≥1, then 𝑟≥√2/2. Let ℎ₁(𝐾)=log𝜂_𝐾(𝑡)−log𝑡 and ℎ₂(𝐾)=𝐾−1/𝐾, then ℎ₁(1)=ℎ₂(1)=0, ℎ(𝐾)=ℎ₁(𝐾)/ℎ₂(𝐾), and ℎ_1(𝐾)ℎ_2(𝐾)=2𝒦^(𝑠)²/𝜇(𝑟)1+𝐾⁻²=2𝜇(𝑟)𝐺(𝐾),(3.6) 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 Theorem 1.2. Differentiating 𝐹_𝑐(𝐾) gives 𝐹^𝑐(𝐾)=log𝜂^𝐾(𝑡)−log𝑡𝐾−1−𝑐+𝐾2𝒦^(𝑠)²(𝐾−1)/𝜇(𝑟)−log𝜂_𝐾(𝑡)−log𝑡=2𝒦_(𝐾−1)²(𝑠)²𝐾(𝐾−1)/𝜇(𝑟)−log𝜂_𝐾(𝑡)−log𝑡(𝐾−1)²−𝑐.(3.7) Let 𝐻(𝐾)=2𝒦^(𝑠)²𝐾(𝐾−1)/𝜇(𝑟)−log𝜂_𝐾(𝑡)−log𝑡(𝐾−1)²,(3.8)𝐻₁(𝐾)=2𝒦^(𝑠)²𝐾(𝐾−1)/𝜇(𝑟)−(log𝜂_𝐾(𝑡)−log𝑡), and 𝐻₂(𝐾)=(𝐾−1)², then 𝐻(𝐾)=𝐻₁(𝐾)/𝐻₂(𝐾), 𝐻₁(1)=𝐻₂(1)=0, and 𝐻_1(𝐾)𝐻_2(𝐾)=𝐻³(𝐾)≡4𝜋𝜇(𝑟)ℰ(𝑠)−𝑠^2𝒦(𝑠)𝒦^(𝑠)³.(3.9)
Clearly, that 𝐻₃(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 lim_𝐾→1𝐻(𝐾)=𝐵(𝑟),lim_𝐾→∞𝐻(𝐾)=𝐴(𝑟).(3.10)
For part (1), if 𝑐>𝐴(𝑟), then from (3.7) and (3.8), we know that 𝐹_𝑐′(𝐾)<0 for 𝐾∈(1,∞) and 𝐹_𝑐(𝐾) is strictly decreasing in (1,∞). Moreover, lim_𝐾→1𝐹_𝑐(𝐾)=4𝒦(𝑟)𝒦^(𝑟)/𝜋−𝑐,lim_𝐾→∞𝐹_𝑐(𝐾)=−∞.(3.11) If 𝑐=𝐴(𝑟), then 𝐹_𝑐(𝐾) is also strictly decreasing in (1,∞) and 𝐹_𝑐(1⁺)=[4𝒦(𝑟)𝒦^(𝑟)/𝜋]−𝐴(𝑟), and from Lemma 2.2(4) we get lim_𝐾→∞𝐹_𝑐(𝐾)=lim_𝐾→∞𝐾=𝐴(𝑟)−𝐾4−lo1g2−−2lloogg𝑡.𝑠^−2𝜇𝑠^+𝐴(𝑟)−log𝑡(3.12)
Therefore, inequalities (1.10) and (1.11) follows from (3.12) and the monotonicity of 𝐹_𝑐(𝐾) when 𝑐=𝐴(𝑟).
For part (2), if 𝑐≤𝐵(𝑟), then that 𝐹_𝑐(𝐾) is strictly increasing in (1,∞) follows from (3.7) and (3.8). Note that lim_𝐾→1𝐹_𝑐(𝐾)=4𝒦(𝑟)𝒦^(𝑟)/𝜋−𝑐,lim_𝐾→∞𝐹_𝑐(𝐾)=+∞.(3.13)
Therefore, inequalities (1.12) and (1.13) follow from (3.13) and the monotonicity of 𝐹_𝑐(𝐾) when 𝑐=𝐵(𝑟).
For part (3), if 𝐵(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 𝐹^𝑐(𝐾)<0 for 𝐾∈(1,𝐾₁) and 𝐹^𝑐(𝐾)>0 for 𝐾∈(𝐾₁,∞). Hence, 𝐹_𝑐(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).

Taking 𝑡=1 in Theorem 1.2, we get the following corollary.

Corollary 3.1. Let 𝑎 and 𝑏 be defined as in Theorem 1.2, 𝑐∈ℝ, and 𝑓_𝑐(𝐾)=𝐾{[log𝜆(𝐾)]/(𝐾−1)−𝑐}. Then,(1)if 𝑐>𝜋, then 𝑓_𝑐(𝐾) is strictly decreasing from (1,∞) onto (−∞,𝑎−𝑐); if 𝑐=𝜋, then 𝑓_𝑐(𝐾) is strictly decreasing from (1,∞) onto (𝜋−4log2,𝑎−𝜋);(2)if 𝑐≤𝑏, then 𝑓_𝑐(𝐾) is strictly increasing from (1,∞) onto (𝑎−𝑐,∞);(3)if 𝑏<𝑐<𝜋, then there exists 𝐾₂∈(1,∞), such that 𝑓_𝑐(𝐾) is strictly decreasing in (1,𝐾₂) and strictly increasing in (𝐾₂,∞);(4)𝑓_𝑐(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. Let 𝑎 and 𝑏 be defined as in Theorem 1.2, then the following inequality max𝑒^{(𝐾−1)(𝜋+((𝜋−4log2)/𝐾))},𝑒^{𝑏(𝐾−(1/𝐾))}<𝜆(𝐾)<𝑒^{(𝐾−1)(𝜋+((𝑎−𝜋)/𝐾))}.(3.14) holds for all 𝐾∈(1,∞).

Acknowledgments

This research was supported by the Natural Science Foundation of China (Grants nos. 11071059, 11071069, and 11171307) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant no. T200924).