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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 671765, 8 pages
http://dx.doi.org/10.1155/2011/671765
Research Article

Monotonicity, Convexity, and Inequalities Involving the Agard Distortion Function

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
3Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 22 June 2011; Accepted 10 November 2011

Academic Editor: Martin D.Β Schechter

Copyright Β© 2011 Yu-Ming Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present some monotonicity, convexity, and inequalities for the Agard distortion function πœ‚πΎ(𝑑) and improve some well-known results.

1. Introduction

For π‘Ÿβˆˆ[0,1], Lengedre’s complete elliptic integrals of the first and second kind [1] are defined by ξ€œπ’¦=𝒦(π‘Ÿ)=0πœ‹/2ξ€·1βˆ’π‘Ÿ2sin2πœƒξ€Έβˆ’1/2π’¦π‘‘πœƒ,ξ…žξ€·π‘Ÿ(π‘Ÿ)=π’¦ξ…žξ€Έπœ‹,𝒦(0)=2ξ€œ,𝒦(1)=∞,(1.1)β„°=β„°(π‘Ÿ)=0πœ‹/2ξ€·1βˆ’π‘Ÿ2sin2πœƒξ€Έ1/2β„°π‘‘πœƒ,ξ…žξ€·π‘Ÿ(π‘Ÿ)=β„°ξ…žξ€Έπœ‹,β„°(0)=2,β„°(1)=1,(1.2) respectively. Here and in what follows, we set π‘Ÿξ…ž=√1βˆ’π‘Ÿ2.

Let πœ‡(π‘Ÿ) be the modulus of the plan GrΓΆtzsch ring 𝐁2⧡[0,π‘Ÿ] for π‘Ÿβˆˆ(0,1), where 𝐁2 is the unit disk. Then, it follows from [2] that πœ‹πœ‡(π‘Ÿ)=2π’¦ξ…ž(π‘Ÿ)𝒦(π‘Ÿ).(1.3)

For 𝐾∈(0,∞), the Hersch-Pfluger distortion function πœ‘πΎ(π‘Ÿ) is defined as πœ‘πΎ(π‘Ÿ)=πœ‡βˆ’1ξ‚΅πœ‡(π‘Ÿ)𝐾forπ‘Ÿβˆˆ(0,1),πœ‘πΎ(0)=πœ‘πΎ(1)βˆ’1=0,(1.4) while the Agard distortion function πœ‚πΎ(𝑑) and the linear distortion function πœ†(𝐾) are defined by πœ‚πΎξ‚Έπœ‘(𝑑)=𝐾(π‘Ÿ)πœ‘1/𝐾(π‘Ÿξ…ž)ξ‚Ή,πœ†(𝐾)=πœ‚πΎξ‚™(1),π‘Ÿ=𝑑1+π‘‘π‘‘βˆˆ(0,∞),(1.5) respectively.

It is well known that the functions πœ‚πΎ(𝑑) and πœ†(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 πœ‚πΎ(𝑑), and in [9–15] the authors established a number of remarkable properties for the Agard distortion function πœ‚πΎ(𝑑).

In [14], the authors proved that π‘’πœ‹(πΎβˆ’1)<πœ†(𝐾)<π‘’π‘Ž(πΎβˆ’1)𝑒,(1.6)𝑏(πΎβˆ’1/𝐾)<πœ†(𝐾)<π‘’πœ‹(πΎβˆ’1/𝐾)(1.7) for all 𝐾∈(1,∞), where βˆšπ‘Ž=(4/πœ‹)𝒦(1/2)2=4.3768…, 𝑏=π‘Ž/2. Recently, Anderson et al. [15] established that πœ†(𝐾)<𝑒(πœ‹+𝑏/𝐾)(πΎβˆ’1),𝑒(1.8)[log2+(π‘Žβˆ’log2)/𝐾](πΎβˆ’1)<πœ†(𝐾)<𝑒[πœ‹+(π‘Žβˆ’log2)/𝐾](πΎβˆ’1)(1.9) for all 𝐾∈(1,∞), where π‘Ž 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 πœ‚πΎ(𝑑) 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,∞), βˆšπ‘Ž=(4/πœ‹)𝒦(1/2)2=4.3768…, 𝑏=π‘Ž/2, and π‘βˆˆβ„. Then, the following statements are true.(1)𝑓(𝐾)=πœ†(𝐾)/𝐾𝑐 is strictly increasing from (1,∞) onto (1,∞) for π‘β‰€π‘Ž; if 𝑐>π‘Ž, then there exists 𝐾0∈(1,∞), such that 𝑓 is strictly decreasing in (1,𝐾0) and strictly increasing in (𝐾0,∞). In particular, the inequality πœ†(𝐾)β‰₯𝐾𝑐 holds for all 𝐾∈(1,∞) with the best possible constant 𝑐=π‘Ž.(2)𝑔(𝐾)=[logπœ‚πΎ(𝑑)βˆ’log𝑑]/(πΎβˆ’1) is convex in (1,∞) for fixed π‘‘βˆˆ(0,∞).(3)If 𝑑β‰₯1 and βˆšπ‘Ÿ=𝑑/(1+𝑑), then β„Ž(𝐾)=[logπœ‚πΎ(𝑑)βˆ’log𝑑]/(πΎβˆ’1/𝐾) is strictly increasing from (1,∞) onto (2𝒦(π‘Ÿ)π’¦ξ…ž(π‘Ÿ)/πœ‹,πœ‹π’¦(π‘Ÿ)/π’¦ξ…ž(π‘Ÿ)).

Theorem 1.2. Let π‘‘βˆˆ(0,∞), βˆšπ‘Ÿ=𝑑/(1+𝑑), βˆšπ‘Ž=(4/πœ‹)𝒦(1/2)2,𝑏=π‘Ž/2, 𝐴(π‘Ÿ)=πœ‹2/(2πœ‡(π‘Ÿ)), 𝐡(π‘Ÿ)=8𝒦(π‘Ÿ)π’¦ξ…ž(π‘Ÿ)2[β„°(π‘Ÿ)βˆ’π‘Ÿξ…ž2𝒦(π‘Ÿ)]/πœ‹2, and 𝐹𝑐(𝐾)=𝐾[(logπœ‚πΎ(𝑑)βˆ’log𝑑)/(πΎβˆ’1)βˆ’π‘]. Then, the following statements are true.(1)𝐹𝑐(𝐾) is strictly decreasing from (1,∞) onto (βˆ’βˆž,4𝒦(π‘Ÿ)𝒦′(π‘Ÿ)/πœ‹βˆ’π‘) for 𝑐>𝐴(π‘Ÿ). If 𝑐=𝐴(π‘Ÿ), then 𝐹𝑐(𝐾) is strictly decreasing from (1,∞) onto (𝐴(π‘Ÿ)βˆ’4log2βˆ’log𝑑,4𝒦(π‘Ÿ)π’¦ξ…ž(π‘Ÿ)/πœ‹βˆ’π΄(π‘Ÿ)). Moreover, 𝑑𝑒(πΎβˆ’1)(𝐴(π‘Ÿ)+((𝐴(π‘Ÿ)βˆ’4log2βˆ’log𝑑)/𝐾))<πœ‚πΎ(𝑑)<𝑑𝑒(πΎβˆ’1)(𝐴(π‘Ÿ)+((4𝒦(π‘Ÿ)𝒦′(π‘Ÿ)/πœ‹βˆ’π΄(π‘Ÿ))/𝐾))(1.10) for all π‘‘βˆˆ(0,∞) and 𝐾∈(1,∞). In particular, if 𝑑=1, then (1.10) becomes 𝑒(πΎβˆ’1)(πœ‹+((πœ‹βˆ’4log2)/𝐾))<πœ†(𝐾)<𝑒(πΎβˆ’1)(πœ‹+((π‘Žβˆ’πœ‹)/𝐾)).(1.11)(2)If 𝑐≀𝐡(π‘Ÿ), then 𝐹𝑐(𝐾) is strictly increasing from (1,∞) onto (4𝒦(π‘Ÿ)π’¦ξ…ž(π‘Ÿ)/πœ‹βˆ’π‘,∞). Moreover, πœ‚πΎ(𝑑)>𝑑𝑒(πΎβˆ’1)(𝐡(π‘Ÿ)+((4𝒦(π‘Ÿ)𝒦′(π‘Ÿ)/πœ‹βˆ’π΅(π‘Ÿ))/𝐾))(1.12) for all π‘‘βˆˆ(0,∞) and 𝐾∈(1,∞). In particular, if 𝑑=1, then (1.12) becomes πœ†(𝐾)>𝑒(πΎβˆ’1)(𝑏+(𝑏/𝐾))=𝑒𝑏(πΎβˆ’(1/𝐾)).(1.13)(3)If 𝐡(π‘Ÿ)<𝑐<𝐴(π‘Ÿ), 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 π‘‘βˆˆ(0,∞),𝐾∈(0,∞), βˆšπ‘Ÿ=𝑑/(1+𝑑)∈(0,1), and 𝑠=πœ‘πΎ(π‘Ÿ). Then, 𝑑𝒦(π‘Ÿ)=π‘‘π‘Ÿβ„°(π‘Ÿ)βˆ’π‘Ÿξ…ž2𝒦(π‘Ÿ)π‘Ÿπ‘Ÿξ…ž2,𝑑ℰ(π‘Ÿ)=π‘‘π‘Ÿβ„°(π‘Ÿ)βˆ’π’¦(π‘Ÿ)π‘Ÿ,𝒦(π‘Ÿ)β„°ξ…ž(π‘Ÿ)+π’¦ξ…ž(π‘Ÿ)β„°(π‘Ÿ)βˆ’π’¦(π‘Ÿ)π’¦ξ…ž(πœ‹π‘Ÿ)=2,π‘‘πœ‡(π‘Ÿ)πœ‹π‘‘π‘Ÿ=βˆ’24π‘Ÿπ‘Ÿξ…ž2𝒦(π‘Ÿ)2,πœ•π‘ =πœ•π‘Ÿπ‘ π‘ ξ…ž2𝒦(𝑠)π’¦ξ…ž(𝑠)π‘Ÿπ‘Ÿξ…ž2𝒦(π‘Ÿ)π’¦ξ…ž,(π‘Ÿ)πœ•π‘ =2πœ•πΎπœ‹πΎπ‘ π‘ ξ…ž2𝒦(𝑠)π’¦ξ…žπœ‘(𝑠),𝐾(π‘Ÿ)2+πœ‘1/πΎξ€·π‘Ÿξ…žξ€Έ2πœ‚=1,𝐾𝑠(𝑑)=π‘ ξ…žξ‚2,πœ•πœ‚πΎ(𝑑)=4πœ•πΎπœ‚πœ‹πΎπΎ(𝑑)𝒦(𝑠)π’¦ξ…ž2(𝑠)=π’¦πœ‡(π‘Ÿ)ξ…ž(𝑠)2πœ‚πΎ(𝑑).(2.1)

Lemma 2.1 (see [14, Theorem  1.25]). For βˆ’βˆž<π‘Ž<𝑏<∞, let 𝑓,π‘”βˆΆ[π‘Ž,𝑏]→ℝ be continuous on [π‘Ž,𝑏] and differentiable on (π‘Ž,𝑏), and let π‘”ξ…ž(π‘₯)β‰ 0 on (π‘Ž,𝑏). If π‘“ξ…ž(π‘₯)/π‘”ξ…ž(π‘₯) is increasing (decreasing) on (π‘Ž,𝑏), then so are 𝑓(π‘₯)βˆ’π‘“(π‘Ž),𝑔(π‘₯)βˆ’π‘”(π‘Ž)𝑓(π‘₯)βˆ’π‘“(𝑏)𝑔(π‘₯)βˆ’π‘”(𝑏).(2.2) If π‘“ξ…ž(π‘₯)/π‘”ξ…ž(π‘₯) 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)  [β„°(π‘Ÿ)βˆ’π‘Ÿβ€²2𝒦(π‘Ÿ)]/π‘Ÿ2 is strictly increasing from (0,1) onto (πœ‹/4,1);
(2)β€‰π‘Ÿβ€²π‘π’¦(π‘Ÿ) is strictly decreasing from (0,1) onto (0,πœ‹/2) if and only if 𝑐β‰₯1/2;
(3) 𝒦(π‘Ÿ)𝒦′(π‘Ÿ) is strictly decreasing in √(0,2/2) and strictly increasing in (√2/2,1);
(4)β€‰β€‰πœ‡(π‘Ÿ)+logπ‘Ÿ is strictly decreasing from (0,1) onto (0,log4).

Lemma 2.3. Let βˆšπ‘Ÿβˆˆ[1/2,1), 𝐾∈(1,∞), and 𝑠=πœ‘πΎ(π‘Ÿ). Then, 𝐺(𝐾)≑{πœ‹/[2𝒦(𝑠)]}2+[πœ‡(π‘Ÿ)/𝒦′(𝑠)]2 is strictly decreasing from (1,∞) onto (𝒦′(π‘Ÿ)2/𝒦(π‘Ÿ)2,πœ‹2/[2𝒦(π‘Ÿ)2]).

Proof. Clearly 𝐺(1+)=πœ‹2/(2𝒦(π‘Ÿ)2), 𝐺(+∞)=π’¦ξ…ž(π‘Ÿ)2/𝒦(π‘Ÿ)2. Differentiating 𝐺(𝐾), one has πΊξ…ž4(𝐾)=πœ‹πΎπœ‡(π‘Ÿ)2𝒦(𝑠)π’¦ξ…ž(𝑠)βˆ’2ξ€Ίβ„°ξ…ž(𝑠)βˆ’π‘ 2π’¦ξ…žξ€»βˆ’πœ‹(𝑠)𝐾𝒦(𝑠)βˆ’2π’¦ξ…žξ‚ƒβ„°(𝑠)(𝑠)βˆ’π‘ ξ…ž2𝒦=4(𝑠)πœ‹πΎπ’¦(𝑠)βˆ’2π’¦ξ…ž(𝑠)βˆ’2𝐺1(𝐾),(2.3) where 𝐺1(𝐾)=[β„°ξ…ž(𝑠)βˆ’π‘ 2π’¦ξ…ž(𝑠)]𝒦(𝑠)3πœ‡(π‘Ÿ)2βˆ’πœ‹2[β„°(𝑠)βˆ’π‘ ξ…ž2𝒦(𝑠)]π’¦ξ…ž(𝑠)3/4.
From Lemma 2.2(1) and (2), we clearly see that 𝐺1(𝐾) is strictly decreasing in (1,∞). Moreover, lim𝐾→1+𝐺1ξ€Ίβ„°(𝐾)=ξ…ž(π‘Ÿ)βˆ’π‘Ÿ2π’¦ξ…žξ€»(π‘Ÿ)𝒦(π‘Ÿ)3πœ‡(π‘Ÿ)2βˆ’πœ‹24ℰ(π‘Ÿ)βˆ’π‘Ÿξ…ž2𝒦𝒦(π‘Ÿ)ξ…ž(π‘Ÿ)3=πœ‹24π’¦ξ…ž(π‘Ÿ)2𝐺2(π‘Ÿ),(2.4) where 𝐺2(π‘Ÿ)=𝒦(π‘Ÿ)[β„°β€²(π‘Ÿ)βˆ’π‘Ÿ2𝒦′(π‘Ÿ)]βˆ’π’¦β€²(π‘Ÿ)[β„°(π‘Ÿ)βˆ’π‘Ÿβ€²2𝒦(π‘Ÿ)] is also strictly decreasing in (0,1). Thus, 𝐺2(π‘Ÿ)≀𝐺2(√2/2)=0 for βˆšπ‘Ÿβˆˆ[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 π‘Ÿ=πœ‡βˆ’1[πœ‹/(2𝐾)] for 𝐾∈(1,∞), then πœ†(𝐾)=(π‘Ÿ/π‘Ÿξ…ž)2, βˆšπ‘Ÿβˆˆ(1/2,1), π‘‘π‘Ÿ=2π‘‘πΎπœ‹π‘Ÿπ‘Ÿξ…ž2π’¦ξ…ž(π‘Ÿ)2,π‘‘πœ†(𝐾)=4π‘‘πΎπœ‹πœ†(𝐾)π’¦ξ…ž(π‘Ÿ)2,(3.1)lim𝐾→+βˆžπ‘“(𝐾)=limπ‘Ÿβ†’1π‘Ÿ2π’¦ξ…ž(π‘Ÿ)π‘Ÿξ…ž2𝒦(π‘Ÿ)=+∞.(3.2) Making use of (3.1), we have 𝐾𝑐+1π‘“ξ…ž(𝐾)πœ†(𝐾)=𝑓14(𝐾)β‰‘πœ‹π’¦ξ…ž(π‘Ÿ)𝒦(π‘Ÿ)βˆ’π‘.(3.3)
It follows from Lemma 2.2(3) that 𝑓1(𝐾) is strictly increasing from (1,∞) onto (π‘Žβˆ’π‘,∞). Then, from (3.2) and (3.3), we know that 𝑓 is strictly increasing from (1,∞) onto (1,∞) for π‘β‰€π‘Ž. If 𝑐>π‘Ž, then there exists 𝐾0∈(1,∞) such that 𝑓 is strictly decreasing in (1,𝐾0) and strictly increasing in (𝐾0,∞). Moreover, the inequality πœ†(𝐾)β‰₯𝐾𝑐 holds for all 𝐾∈(1,∞) with the best possible constant 𝑐=π‘Ž.
For part (2), denote βˆšπ‘Ÿ=𝑑/(1+𝑑). Differentiating 𝑔(𝐾), we get π‘”ξ…ž(𝐾)=2π’¦ξ…ž(𝑠)2ξ€·(πΎβˆ’1)/πœ‡(π‘Ÿ)βˆ’logπœ‚πΎξ€Έ(𝑑)βˆ’log𝑑(πΎβˆ’1)2.(3.4)
Let 𝑔1(𝐾)=2π’¦ξ…ž(𝑠)2(πΎβˆ’1)/πœ‡(π‘Ÿ)βˆ’(logπœ‚πΎ(𝑑)βˆ’log𝑑) and 𝑔2(𝐾)=(πΎβˆ’1)2, then 𝑔1(1)=𝑔2(1)=0, 𝑔′(𝐾)=𝑔1(𝐾)/𝑔2(𝐾) and 𝑔1ξ…ž(𝐾)𝑔2ξ…ž(𝐾)=𝑔32(𝐾)β‰‘βˆ’πœ‡(π‘Ÿ)2ξ€Ίβ„°ξ…ž(𝑠)βˆ’π‘ 2π’¦ξ…žξ€»π’¦(𝑠)ξ…ž(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 𝑑β‰₯1, then βˆšπ‘Ÿβ‰₯2/2. Let β„Ž1(𝐾)=logπœ‚πΎ(𝑑)βˆ’log𝑑 and β„Ž2(𝐾)=πΎβˆ’1/𝐾, then β„Ž1(1)=β„Ž2(1)=0, β„Ž(𝐾)=β„Ž1(𝐾)/β„Ž2(𝐾), and β„Ž1ξ…ž(𝐾)β„Ž2ξ…ž=(𝐾)2π’¦ξ…ž(𝑠)2/πœ‡(π‘Ÿ)1+πΎβˆ’2=2πœ‡(π‘Ÿ)𝐺(𝐾),(3.6) where 𝐺(𝐾) is defined as in Lemma 2.2.
Therefore, β„Ž(𝐾) is strictly increasing in (1,∞) for 𝑑β‰₯1 follows from Lemmas 2.1 and 2.2 together with (3.6). Moreover, making use of l’HΓ΄pital’s rule, we have β„Ž(1+)=2𝒦(π‘Ÿ)π’¦ξ…ž(π‘Ÿ)/πœ‹, β„Ž(∞)=πœ‹π’¦(π‘Ÿ)/π’¦ξ…ž(π‘Ÿ).

Proof of Theorem 1.2. Differentiating 𝐹𝑐(𝐾) gives πΉξ…žπ‘(𝐾)=logπœ‚πΎ(𝑑)βˆ’logπ‘‘ξƒ¬ξ€·πΎβˆ’1βˆ’π‘+𝐾2π’¦ξ…ž(𝑠)2ξ€Έξ€·(πΎβˆ’1)/πœ‡(π‘Ÿ)βˆ’logπœ‚πΎξ€Έ(𝑑)βˆ’log𝑑(πΎβˆ’1)2ξƒ­=2π’¦ξ…ž(𝑠)2𝐾(πΎβˆ’1)/πœ‡(π‘Ÿ)βˆ’logπœ‚πΎξ€Έ(𝑑)βˆ’log𝑑(πΎβˆ’1)2βˆ’π‘.(3.7) Let 𝐻(𝐾)=2π’¦ξ…ž(𝑠)2ξ€»βˆ’ξ€ΊπΎ(πΎβˆ’1)/πœ‡(π‘Ÿ)logπœ‚πΎξ€»(𝑑)βˆ’log𝑑(πΎβˆ’1)2,(3.8)𝐻1(𝐾)=2π’¦ξ…ž(𝑠)2𝐾(πΎβˆ’1)/πœ‡(π‘Ÿ)βˆ’(logπœ‚πΎ(𝑑)βˆ’log𝑑), and 𝐻2(𝐾)=(πΎβˆ’1)2, then 𝐻(𝐾)=𝐻1(𝐾)/𝐻2(𝐾), 𝐻1(1)=𝐻2(1)=0, and 𝐻1ξ…ž(𝐾)𝐻2ξ…ž(𝐾)=𝐻34(𝐾)β‰‘ξ‚ƒβ„°πœ‹πœ‡(π‘Ÿ)(𝑠)βˆ’π‘ ξ…ž2𝒦𝒦(𝑠)ξ…ž(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’HΓ΄pital’s rule leads to lim𝐾→1𝐻(𝐾)=𝐡(π‘Ÿ),limπΎβ†’βˆžπ»(𝐾)=𝐴(π‘Ÿ).(3.10)
For part (1), if 𝑐>𝐴(π‘Ÿ), 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, lim𝐾→1𝐹𝑐(𝐾)=4𝒦(π‘Ÿ)π’¦ξ…žξ€»(π‘Ÿ)/πœ‹βˆ’π‘,limπΎβ†’βˆžπΉπ‘(𝐾)=βˆ’βˆž.(3.11) If 𝑐=𝐴(π‘Ÿ), then 𝐹𝑐(𝐾) is also strictly decreasing in (1,∞) and 𝐹𝑐(1+)=[4𝒦(π‘Ÿ)π’¦ξ…ž(π‘Ÿ)/πœ‹]βˆ’π΄(π‘Ÿ), and from Lemma 2.2(4) we get limπΎβ†’βˆžπΉπ‘(𝐾)=limπΎβ†’βˆžπΎξ€Ίξ€·π‘ πΎβˆ’1βˆ’2logξ…žξ€Έξ€·π‘ βˆ’2πœ‡ξ…žξ€Έξ€»+𝐴(π‘Ÿ)βˆ’log𝑑=𝐴(π‘Ÿ)βˆ’4log2βˆ’log𝑑.(3.12)
Therefore, inequalities (1.10) and (1.11) follows from (3.12) and the monotonicity of 𝐹𝑐(𝐾) when 𝑐=𝐴(π‘Ÿ).
For part (2), if 𝑐≀𝐡(π‘Ÿ), then that 𝐹𝑐(𝐾) is strictly increasing in (1,∞) follows from (3.7) and (3.8). Note that lim𝐾→1𝐹𝑐(𝐾)=4𝒦(π‘Ÿ)π’¦ξ…žξ€»(π‘Ÿ)/πœ‹βˆ’π‘,limπΎβ†’βˆžπΉπ‘(𝐾)=+∞.(3.13)
Therefore, inequalities (1.12) and (1.13) follow from (3.13) and the monotonicity of 𝐹𝑐(𝐾) when 𝑐=𝐡(π‘Ÿ).
For part (3), if 𝐡(π‘Ÿ)<𝑐<𝐴(π‘Ÿ), then from (3.7) and (3.8) together with the monotonicity of 𝐻(𝐾) we clearly see that there exists 𝐾1∈(1,∞), such that πΉξ…žπ‘(𝐾)<0 for 𝐾∈(1,𝐾1) and πΉξ…žπ‘(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).

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

Corollary 3.1. Let π‘Ž and 𝑏 be defined as in Theorem 1.2, π‘βˆˆβ„, and 𝑓𝑐(𝐾)=𝐾{[logπœ†(𝐾)]/(πΎβˆ’1)βˆ’π‘}. Then,(1)if 𝑐>πœ‹, then 𝑓𝑐(𝐾) is strictly decreasing from (1,∞) onto (βˆ’βˆž,π‘Žβˆ’π‘); if 𝑐=πœ‹, then 𝑓𝑐(𝐾) is strictly decreasing from (1,∞) onto (πœ‹βˆ’4log2,π‘Žβˆ’πœ‹);(2)if 𝑐≀𝑏, then 𝑓𝑐(𝐾) is strictly increasing from (1,∞) onto (π‘Žβˆ’π‘,∞);(3)if 𝑏<𝑐<πœ‹, 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. Let π‘Ž and 𝑏 be defined as in Theorem 1.2, then the following inequality 𝑒max(πΎβˆ’1)(πœ‹+((πœ‹βˆ’4log2)/𝐾)),𝑒𝑏(πΎβˆ’(1/𝐾))ξ€Ύ<πœ†(𝐾)<𝑒(πΎβˆ’1)(πœ‹+((π‘Žβˆ’πœ‹)/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).

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