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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 605259, 8 pages
http://dx.doi.org/10.1155/2011/605259
Research Article

Sharp Generalized Seiffert Mean Bounds for Toader Mean

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 4 June 2011; Revised 10 August 2011; Accepted 11 August 2011

Academic Editor: DetlevΒ Buchholz

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

For π‘βˆˆ[0,1], the generalized Seiffert mean of two positive numbers π‘Ž and 𝑏 is defined by 𝑆𝑝(π‘Ž,𝑏)=𝑝(π‘Žβˆ’π‘)/arctan[2𝑝(π‘Žβˆ’π‘)/(π‘Ž+𝑏)],0<𝑝≀1,π‘Žβ‰ π‘;(π‘Ž+𝑏)/2,𝑝=0,π‘Žβ‰ π‘;π‘Ž,π‘Ž=𝑏. In this paper, we find the greatest value 𝛼 and least value 𝛽 such that the double inequality 𝑆𝛼(π‘Ž,𝑏)<𝑇(π‘Ž,𝑏)<𝑆𝛽(π‘Ž,𝑏) holds for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘, and give new bounds for the complete elliptic integrals of the second kind. Here, βˆ«π‘‡(π‘Ž,𝑏)=(2/πœ‹)0πœ‹/2βˆšπ‘Ž2cos2πœƒ+𝑏2sin2πœƒπ‘‘πœƒ denotes the Toader mean of two positive numbers π‘Ž and 𝑏.

1. Introduction

For π‘βˆˆ[0,1], the generalized Seiffert mean of two positive numbers π‘Ž and 𝑏 is defined by π‘†π‘βŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩(π‘Ž,𝑏)=𝑝(π‘Žβˆ’π‘)[]arctan2𝑝(π‘Žβˆ’π‘)/(π‘Ž+𝑏),0<𝑝≀1,π‘Žβ‰ π‘,π‘Ž+𝑏2,𝑝=0,π‘Žβ‰ π‘,π‘Ž,π‘Ž=𝑏.(1.1)

It is well known that 𝑆𝑝(π‘Ž,𝑏) is continuous and strictly increasing with respect to π‘βˆˆ[0,1] for fixed π‘Ž,𝑏>0 with π‘Žβ‰ π‘. In particular, if 𝑝=1/2, then the generalized Seiffert mean reduces to the Seiffert mean 𝑆(π‘Ž,𝑏)=π‘Žβˆ’π‘2arctan((π‘Žβˆ’π‘)/(π‘Ž+𝑏)),π‘Žβ‰ π‘,π‘Ž,π‘Ž=𝑏.(1.2)

Recently, the Seiffert mean and its generalization have been the subject of intensive research, many remarkable inequalities for these means can be found in the literature [1–5].

In [6], Toader introduced the Toader mean 𝑇(π‘Ž,𝑏) of two positive numbers π‘Ž and 𝑏 as follows: 2𝑇(π‘Ž,𝑏)=πœ‹ξ€œ0πœ‹/2βˆšπ‘Ž2cos2πœƒ+𝑏2sin2=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©ξ‚€βˆšπœƒπ‘‘πœƒ,2π‘Žβ„°1βˆ’(𝑏/π‘Ž)2ξ‚πœ‹ξ‚€βˆš,π‘Ž>𝑏,2𝑏ℰ1βˆ’(π‘Ž/𝑏)2ξ‚πœ‹,π‘Ž<𝑏,π‘Ž,π‘Ž=𝑏,(1.3) where βˆ«β„°(π‘Ÿ)=0πœ‹/2(1βˆ’π‘Ÿ2sin2𝑑)1/2𝑑𝑑, π‘Ÿβˆˆ[0,1) is the complete elliptic integral of the second kind.

Vuorinen [7] conjectured that 𝑀3/2(π‘Ž,𝑏)<𝑇(π‘Ž,𝑏)(1.4) for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘, where π‘€π‘βŽ§βŽͺ⎨βŽͺβŽ©ξ‚΅π‘Ž(π‘Ž,𝑏)=𝑝+𝑏𝑝2ξ‚Ά1/π‘βˆš,𝑝≠0,π‘Žπ‘,𝑝=0(1.5) is the power mean of order 𝑝 of two positive numbers π‘Ž and 𝑏. This conjecture was proved by Barnard et al. [8].

In [9], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows:𝑇(π‘Ž,𝑏)<𝑀log2/log(πœ‹/2)(π‘Ž,𝑏)(1.6) for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘.

The main purpose of this paper is to find the greatest value 𝛼 and least value 𝛽 such that the double inequality 𝑆𝛼(π‘Ž,𝑏)<𝑇(π‘Ž,𝑏)<𝑆𝛽(π‘Ž,𝑏) holds for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘ and give new bounds for the complete elliptic integrals of the second kind.

2. Lemmas

In order to establish our main result, we need several formulas and lemmas, which we present in this section.

The following formulas were presented in [10, Appendix E, pages 474-475]: Let π‘Ÿβˆˆ[0,1), then ξ€œπ’¦(π‘Ÿ)=0πœ‹/2ξ€·1βˆ’π‘Ÿ2sin2π‘‘ξ€Έβˆ’1/2πœ‹π‘‘π‘‘,𝒦(0)=2,𝒦(1βˆ’β„°ξ€œ)=+∞,(π‘Ÿ)=0πœ‹/2ξ€·1βˆ’π‘Ÿ2sin2𝑑1/2𝑑𝑑,β„°(0)=πœ‹/2,β„°(1βˆ’)=1,𝑑𝒦(π‘Ÿ)=ξ€·π‘‘π‘Ÿβ„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)π‘Ÿξ€·1βˆ’π‘Ÿ2ξ€Έ,𝑑ℰ(π‘Ÿ)=β„°π‘‘π‘Ÿ(π‘Ÿ)βˆ’π’¦(π‘Ÿ)π‘Ÿ,𝑑ℰ(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)𝑑[]π‘‘π‘Ÿ=π‘Ÿπ’¦(π‘Ÿ),𝒦(π‘Ÿ)βˆ’β„°(π‘Ÿ)=π‘‘π‘Ÿπ‘Ÿβ„°(π‘Ÿ)1βˆ’π‘Ÿ2,ℰ2βˆšπ‘Ÿξƒͺ=ξ€·1+π‘Ÿ2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ).1+π‘Ÿ(2.1)

Lemma 2.1 (see [10, Theorem 1.25]). For βˆ’βˆž<π‘Ž<𝑏<∞, let 𝑓(π‘₯),𝑔(π‘₯)∢[π‘Ž,𝑏]→ℝ be continuous on [π‘Ž,𝑏] and be differentiable on (π‘Ž,𝑏), let π‘”ξ…ž(π‘₯)β‰ 0 for all π‘₯∈(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.

Lemma 2.2. (1)  [β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ)]/π‘Ÿ2 is strictly increasing from (0,1) onto (πœ‹/4,1);
(2)    {[β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ)]/π‘Ÿ2βˆ’πœ‹/4}/π‘Ÿ2 is strictly increasing from (0,1) onto (πœ‹/32,1βˆ’πœ‹/4);
(3)    [𝒦(π‘Ÿ)βˆ’β„°(π‘Ÿ)]/π‘Ÿ2 is strictly increasing from (0,1) onto (πœ‹/4,+∞);
(4)    2β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ) is strictly increasing from (0,1) onto (πœ‹/2,2);
(5)    𝐹(π‘Ÿ)=[(2βˆ’π‘Ÿ2)𝒦(π‘Ÿ)βˆ’2β„°(π‘Ÿ)]/π‘Ÿ4 is strictly increasing from (0,1) onto (πœ‹/16,+∞);
(6)    𝐺(π‘Ÿ)=[4πœ‹βˆ’πœ‹π‘Ÿ2βˆ’8β„°(π‘Ÿ)]/π‘Ÿ4 is strictly increasing from (0,1) onto (3πœ‹/16,3πœ‹βˆ’8).

Proof. Parts (1)–(4) can be found in [10, Theorem 3.21(1), Theorem  3.31(6), and Exercise 3.43(11) and (13)].
For part (5), clearly 𝐹(1βˆ’)=+∞. Let 𝐹1(π‘Ÿ)=(2βˆ’π‘Ÿ2)𝒦(π‘Ÿ)βˆ’2β„°(π‘Ÿ) and 𝐹2(π‘Ÿ)=π‘Ÿ4, then 𝐹(π‘Ÿ)=𝐹1(π‘Ÿ)/𝐹2(π‘Ÿ), 𝐹1(0)=𝐹2(0)=0 and πΉξ…ž1(π‘Ÿ)πΉξ…ž2=ξ€·(π‘Ÿ)β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦4π‘Ÿ2ξ€·1βˆ’π‘Ÿ2ξ€Έ.(2.3)
It follows from (2.3) and part (1) together with Lemma 2.1 that 𝐹(π‘Ÿ) is strictly increasing in (0,1) and 𝐹(0+)=πœ‹/16.
For part (6), clearly 𝐺(1βˆ’)=3πœ‹βˆ’8. Let 𝐺1(π‘Ÿ)=4πœ‹βˆ’πœ‹π‘Ÿ2βˆ’8β„°(π‘Ÿ) and 𝐺2(π‘Ÿ)=π‘Ÿ4, then 𝐺(π‘Ÿ)=𝐺1(π‘Ÿ)/𝐺2(π‘Ÿ), 𝐺1(0)=𝐺2(0)=0, and πΊξ…ž1(π‘Ÿ)2πΊξ…ž2=ξ€·(π‘Ÿ)2βˆ’π‘Ÿ2𝒦(π‘Ÿ)βˆ’2β„°(π‘Ÿ)π‘Ÿ4+ξ€Ίξ€·β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)/π‘Ÿ2βˆ’πœ‹/4π‘Ÿ2.(2.4)
From (2.4), parts (2) and (5) together with Lemma 2.1, we know that 𝐺(π‘Ÿ) is strictly increasing in (0,1), and 𝑓(0+)=3πœ‹/16.

Lemma 2.3. (1)βˆšπ‘”(π‘Ÿ)=arctan(√3π‘Ÿ/2)βˆ’3πœ‹π‘Ÿ/{4[2β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ)]} is strictly increasing from (0,1) onto √(0,arctan(√3/2)βˆ’3πœ‹/8).
(2)𝑓(π‘Ÿ)=arctanπ‘Ÿβˆ’πœ‹π‘Ÿ/{2[2β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ)]}<0 for π‘Ÿβˆˆ(0,1).

Proof. For part (1), clearly 𝑔(0+)=0 and 𝑔(1βˆ’βˆš)=arctan(√3/2)βˆ’3πœ‹/8=0.0335β‹―>0. Simple computation leads to π‘”ξ…ž2√(π‘Ÿ)=34+3π‘Ÿ2βˆ’βˆš3πœ‹β„°(π‘Ÿ)4ξ€Ίξ€·2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)2=√3π‘Ÿ4β„°(π‘Ÿ)4ξ€·4+3π‘Ÿ2ξ€·ξ€Έξ€Ί2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)2𝑔1(π‘Ÿ),(2.5) where 𝑔1(π‘Ÿ)={8[2β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ)]2βˆ’πœ‹(4+3π‘Ÿ2)β„°(π‘Ÿ)}/[π‘Ÿ4β„°(π‘Ÿ)].
Making use of Lemma 2.2 (1), (2), and (6), we get 𝑔18(π‘Ÿ)=ℰ⋅ℰ(π‘Ÿ)(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)π‘Ÿ2ξƒ­2+β„°ξ€·16ξ€½ξ€Ί(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)/π‘Ÿ2ξ€Ύβˆ’πœ‹/4π‘Ÿ2βˆ’4πœ‹βˆ’πœ‹π‘Ÿ2βˆ’8β„°(π‘Ÿ)π‘Ÿ4>16πœ‹β‹…ξ‚€πœ‹42πœ‹+16β‹…32βˆ’(3πœ‹βˆ’8)=8βˆ’3πœ‹2>0.(2.6)
Therefore, part (1) follows from (2.5) and (2.6) together with the limiting values of 𝑔(π‘Ÿ) at π‘Ÿ=0 and π‘Ÿ=1.
For part (2), simple computations yield that limπ‘Ÿβ†’0+𝑓(π‘Ÿ)=limπ‘Ÿβ†’1βˆ’π‘“(π‘Ÿ)=0,(2.7)𝑓𝑓′(π‘Ÿ)=1(π‘Ÿ)2ξ€·1+π‘Ÿ2ξ€·ξ€Έξ€Ί2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)2,(2.8) where 𝑓1(π‘Ÿ)=2[2β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ)]2βˆ’πœ‹(1+π‘Ÿ2)β„°(π‘Ÿ). Note that limπ‘Ÿβ†’0+𝑓1(π‘Ÿ)=0,(2.9)limπ‘Ÿβ†’1βˆ’π‘“1𝑓(π‘Ÿ)=8βˆ’2πœ‹>0,(2.10)ξ…ž14ξ€Ίξ€·(π‘Ÿ)=2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)ξ€»ξ€Ίβ„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)π‘Ÿξ€·βˆ’2πœ‹π‘Ÿβ„°(π‘Ÿ)βˆ’πœ‹1+π‘Ÿ2ξ€Έβ„°(π‘Ÿ)βˆ’π’¦(π‘Ÿ)π‘Ÿ=π‘Ÿπ‘“2(π‘Ÿ),(2.11) where 𝑓2(π‘Ÿ)=4[2β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ)][β„°(π‘Ÿ)βˆ’(1βˆ’π‘Ÿ2)𝒦(π‘Ÿ)]/π‘Ÿ2βˆ’2πœ‹β„°(π‘Ÿ)+πœ‹(1+π‘Ÿ2)[𝒦(π‘Ÿ)βˆ’β„°(π‘Ÿ)]/π‘Ÿ2.
From Lemma 2.2(1), (3), and (4) together with the monotonicity of β„°(π‘Ÿ) we know that 𝑓2(π‘Ÿ) is strictly increasing in (0,1). Moreover, limπ‘Ÿβ†’0+𝑓2πœ‹(π‘Ÿ)=βˆ’24,(2.12)limπ‘Ÿβ†’1βˆ’π‘“2(π‘Ÿ)=+∞.(2.13)
Equations (2.11)–(2.13) and the monotonicity of 𝑓2(π‘Ÿ) lead to the conclusion that there exists π‘Ÿ0∈(0,1) such that 𝑓1(π‘Ÿ) is strictly decreasing in (0,π‘Ÿ0) and strictly increasing in (π‘Ÿ0,1).
It follows from (2.8)–(2.10) and the piecewise monotonicity of 𝑓1(π‘Ÿ) that there exists π‘Ÿ1∈(0,1) such that 𝑓(π‘Ÿ) is strictly decreasing in (0,π‘Ÿ1) and strictly increasing in (π‘Ÿ1,1).
Therefore, part (2) follows from (2.7) and the piecewise monotonicity of 𝑓(π‘Ÿ).

3. Main Result

Theorem 3.1. Inequality π‘†βˆš3/4(π‘Ž,𝑏)<𝑇(π‘Ž,𝑏)<𝑆1/2(π‘Ž,𝑏) holds for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘, and π‘†βˆš3/4(π‘Ž,𝑏) and 𝑆1/2(π‘Ž,𝑏) are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean 𝑇(π‘Ž,𝑏), respectively.

Proof. Firstly, we prove that π‘†βˆš3/4(π‘Ž,𝑏)<𝑇(π‘Ž,𝑏)<𝑆1/2(π‘Ž,𝑏)(3.1) for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘.
Without loss of generality, we assume that π‘Ž>𝑏. Let 𝑑=𝑏/π‘Ž<1, π‘Ÿ=(1βˆ’π‘‘)/(1+𝑑). Then (1.1) and (1.3) lead to 𝑇(π‘Ž,𝑏)βˆ’π‘†βˆš3/4(π‘Ž,𝑏)=2π‘Žπœ‹β„°ξ‚€βˆš1βˆ’π‘‘2ξ‚βˆ’βˆš3π‘Ž(1βˆ’π‘‘)ξ‚ƒβˆš4arctan3ξ‚„=(1βˆ’π‘‘)/2(1+𝑑)2π‘Žπœ‹β„°ξƒ©2βˆšπ‘Ÿξƒͺβˆ’βˆš1+π‘Ÿ3ar√2(1+π‘Ÿ)arctanξ‚€ξ‚€ξ‚π‘Ÿξ‚=3/22π‘Žπœ‹ξ€Ίξ€·2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)βˆ’βˆš1+π‘Ÿ3ar√2(1+π‘Ÿ)arctanξ‚€ξ‚€ξ‚π‘Ÿξ‚=ξ€Ίξ€·3/22π‘Ž2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)βˆšπœ‹(1+π‘Ÿ)arctanξ‚€ξ‚€ξ‚π‘Ÿξ‚3/2𝑔(π‘Ÿ),(3.2)𝑇(π‘Ž,𝑏)βˆ’π‘†1/2(π‘Ž,𝑏)=2π‘Žπœ‹β„°ξ‚€βˆš1βˆ’π‘‘2ξ‚βˆ’π‘Ž(1βˆ’π‘‘)=2arctan((1βˆ’π‘‘)/(1+𝑑))2π‘Žπœ‹β„°ξƒ©2βˆšπ‘Ÿξƒͺβˆ’1+π‘Ÿπ‘Žπ‘Ÿ=(1+π‘Ÿ)arctanπ‘Ÿ2π‘Žπœ‹ξ€Ίξ€·2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)βˆ’1+π‘Ÿar=ξ€Ίξ€·(1+π‘Ÿ)arctanπ‘Ÿ2π‘Ž2β„°(π‘Ÿ)βˆ’1βˆ’π‘Ÿ2𝒦(π‘Ÿ)πœ‹(1+π‘Ÿ)arctanπ‘Ÿπ‘“(π‘Ÿ),(3.3) where 𝑔(π‘Ÿ) and 𝑓(π‘Ÿ) are defined as in Lemma 2.3.
Therefore, inequality (3.1) follows from (3.2) and (3.3) together with Lemma 2.3.
Next, we prove that π‘†βˆš3/4(π‘Ž,𝑏) and 𝑆1/2(π‘Ž,𝑏) are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean 𝑇(π‘Ž,𝑏), respectively.
For any πœ€>0 and 0<π‘₯<1, from (1.1) and (1.3) one has limπ‘₯β†’0𝑆1/2βˆ’πœ€ξ€»=(1,π‘₯)βˆ’π‘‡(1,π‘₯)1βˆ’2πœ€βˆ’22arctan(1βˆ’2πœ€)πœ‹<1βˆ’22arctan1πœ‹π‘†=0,(3.4)√3/4+πœ€(1,1βˆ’π‘₯)βˆ’π‘‡(1,1βˆ’π‘₯)=𝐽(π‘₯)√arctan,3+4πœ€π‘₯/2(2βˆ’π‘₯)(3.5) where √𝐽(π‘₯)=(√3/4+πœ€)π‘₯βˆ’2β„°(2π‘₯βˆ’π‘₯2√)arctan{[(3+4πœ€)π‘₯]/[2(2βˆ’π‘₯)]}/πœ‹.
Letting π‘₯β†’0 and making use of Taylor expansion, we get ξƒ©βˆšπ½(π‘₯)=34ξƒͺξƒ©βˆš+πœ€π‘₯βˆ’34ξƒͺπ‘₯1+πœ€1βˆ’21π‘₯+π‘₯162ξ€·π‘₯+π‘œ2ξ€Έξ‚„Γ—βŽ§βŽͺ⎨βŽͺ⎩11+2⎑⎒⎒⎣1π‘₯+4βˆ’13ξƒ©βˆš34ξƒͺ+πœ€2⎀βŽ₯βŽ₯⎦π‘₯2ξ€·π‘₯+π‘œ2ξ€ΈβŽ«βŽͺ⎬βŽͺ⎭=πœ€3ξƒ©βˆš32√+πœ€ξƒͺ34ξƒͺπ‘₯+πœ€3ξ€·π‘₯+π‘œ3ξ€Έ.(3.6)
Inequality (3.4) and equations (3.5) and (3.6) imply that for any πœ€>0 there exist 𝛿1=𝛿1(πœ€)>0 and 𝛿2=𝛿2(πœ€)>0, such that π‘†βˆš3/4+πœ€(1,1βˆ’π‘₯)>𝑇(1,1βˆ’π‘₯) for π‘₯∈(0,𝛿1) and 𝑆1/2βˆ’πœ€(1,π‘₯)<𝑇(1,π‘₯) for π‘₯∈(0,𝛿2).

From Theorem 3.1, we get new bounds for the complete elliptic integrals of the second kind as follows.

Corollary 3.2. The inequality βˆšξ‚€βˆš3πœ‹1βˆ’1βˆ’π‘Ÿ2ξ‚ξ‚†βˆš8arctan3ξ‚€βˆš1βˆ’1βˆ’π‘Ÿ2/2ξ‚€βˆš1+1βˆ’π‘Ÿ2πœ‹ξ‚€βˆšξ‚ξ‚„ξ‚‡<𝐸(π‘Ÿ)<1βˆ’1βˆ’π‘Ÿ2ξ‚βˆš4arctan1βˆ’1βˆ’π‘Ÿ2/ξ‚€βˆš1+1βˆ’π‘Ÿ2(3.7) holds for all π‘Ÿβˆˆ(0,1).

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grant 11071069 and Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924. The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions.

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