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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 684834, 7 pages
http://dx.doi.org/10.1155/2012/684834
Research Article

A Sharp Double Inequality between Seiffert, Arithmetic, and Geometric Means

1College of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China
2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 2 July 2012; Accepted 21 August 2012

Academic Editor: JosefΒ DiblΓ­k

Copyright Β© 2012 Wei-Ming Gong 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 fixed 𝑠β‰₯1 and any 𝑑1,𝑑2∈(0,1/2) we prove that the double inequality 𝐺𝑠(𝑑1π‘Ž+(1βˆ’π‘‘1)𝑏,𝑑1𝑏+(1βˆ’π‘‘1)π‘Ž)𝐴1βˆ’π‘ (π‘Ž,𝑏)<𝑃(π‘Ž,𝑏)<𝐺𝑠(𝑑2π‘Ž+(1βˆ’π‘‘2)𝑏,𝑑2𝑏+(1βˆ’π‘‘2)π‘Ž)𝐴1βˆ’π‘ (π‘Ž,𝑏) holds for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘ if and only if 𝑑1βˆšβ‰€(1βˆ’1βˆ’(2/πœ‹)2/𝑠)/2 and 𝑑2√β‰₯(1βˆ’1/3𝑠)/2. Here, 𝑃(π‘Ž,𝑏), 𝐴(π‘Ž,𝑏) and 𝐺(π‘Ž,𝑏) denote the Seiffert, arithmetic, and geometric means of two positive numbers π‘Ž and 𝑏, respectively.

1. Introduction

The Seiffert mean 𝑃(π‘Ž,𝑏) [1] of two distinct positive numbers π‘Ž and 𝑏 is defined by 𝑃(π‘Ž,𝑏)=π‘Žβˆ’π‘ξ‚€βˆš4arctan.π‘Ž/π‘βˆ’πœ‹(1.1)

Recently, the Seiffert mean 𝑃(π‘Ž,𝑏) has been the subject of intensive research. In particular, many remarkable inequalities for 𝑃(π‘Ž,𝑏) can be found in the literature [2–17]. The Seiffert mean 𝑃(π‘Ž,𝑏) can be rewritten as (see [6, (2.4)]) 𝑃(π‘Ž,𝑏)=π‘Žβˆ’π‘.2arcsin((π‘Žβˆ’π‘)/(π‘Ž+𝑏))(1.2)

Let 𝐴(π‘Ž,𝑏)=(π‘Ž+𝑏)/2, √𝐺(π‘Ž,𝑏)=π‘Žπ‘ and 𝐻(π‘Ž,𝑏)=2π‘Žπ‘/(π‘Ž+𝑏) be the classical arithmetic, geometric, and harmonic means of two positive numbers π‘Ž and 𝑏, respectively. Then it is well known that inequalities 𝐻(π‘Ž,𝑏)<𝐺(π‘Ž,𝑏)<𝑃(π‘Ž,𝑏)<𝐴(π‘Ž,𝑏) hold for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘.

For 𝛼,𝛽,πœ†,πœ‡βˆˆ(0,1/2), Chu et al. [18, 19] proved that the double inequalities 𝐻𝐺(π›Όπ‘Ž+(1βˆ’π›Ό)𝑏,𝛼𝑏+(1βˆ’π›Ό)π‘Ž)<𝑃(π‘Ž,𝑏)<𝐺(π›½π‘Ž+(1βˆ’π›½)𝑏,𝛽𝑏+(1βˆ’π›½)π‘Ž),(πœ†π‘Ž+(1βˆ’πœ†)𝑏,πœ†π‘+(1βˆ’πœ†)π‘Ž)<𝑃(π‘Ž,𝑏)<𝐻(πœ‡π‘Ž+(1βˆ’πœ‡)𝑏,πœ‡π‘+(1βˆ’πœ‡)π‘Ž)(1.3) hold for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘ if and only if βˆšπ›Όβ‰€(1βˆ’1βˆ’4/πœ‹2)/2, βˆšπ›½β‰₯(3βˆ’3)/6, βˆšπœ†β‰€(1βˆ’1βˆ’2/πœ‹)/2 and βˆšπœ‡β‰₯(6βˆ’6)/12.

Let π‘‘βˆˆ(0,1/2), 𝑠β‰₯1 and 𝑄𝑑,𝑠(π‘Ž,𝑏)=𝐺𝑠(π‘‘π‘Ž+(1βˆ’π‘‘)𝑏,𝑑𝑏+(1βˆ’π‘‘)π‘Ž)𝐴1βˆ’π‘ (π‘Ž,𝑏),(1.4) then it is not difficult to verify that 𝑄𝑑,1𝑄(π‘Ž,𝑏)=𝐺(π‘‘π‘Ž+(1βˆ’π‘‘)𝑏,𝑑𝑏+(1βˆ’π‘‘)π‘Ž),𝑑,2(π‘Ž,𝑏)=𝐻(π‘‘π‘Ž+(1βˆ’π‘‘)𝑏,𝑑𝑏+(1βˆ’π‘‘)π‘Ž)(1.5) and 𝑄𝑑,𝑠(π‘Ž,𝑏) is strictly increasing with respect to π‘‘βˆˆ(0,1/2) for fixed π‘Ž,𝑏>0 with π‘Žβ‰ π‘.

It is natural to ask what are the largest value 𝑑1=𝑑1(𝑠) and the least value 𝑑2=𝑑2(𝑠) in (0,1/2) such that the double inequality 𝑄𝑑1,𝑠(π‘Ž,𝑏)<𝑃(π‘Ž,𝑏)<𝑄𝑑2,𝑠(π‘Ž,𝑏) holds for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘ and 𝑠β‰₯1. The main purpose of this paper is to answer this question.

2. Main Result

In order to establish our main result we need two lemmas, which we present in the following.

Lemma 2.1. If 𝑠β‰₯1, then 1/(3𝑠)+(2/πœ‹)2/𝑠<1.

Proof. Consider the following: 1𝑓(𝑠)=+ξ‚€23π‘ πœ‹ξ‚2/𝑠.(2.1)
Then simple computations lead to lim𝑠→+βˆžπ‘“π‘“(𝑠)=1,(2.2)ξ…ž2(𝑠)=𝑠2πœ‹log2ξ‚Έξ‚€2πœ‹ξ‚2/π‘ βˆ’1ξ‚Ήβ‰₯26log(πœ‹/2)𝑠2πœ‹log2ξ‚Έξ‚€2πœ‹ξ‚2βˆ’1ξ‚Ή=6log(πœ‹/2)24log(πœ‹/2)βˆ’πœ‹23πœ‹2𝑠2(2.3) for 𝑠β‰₯1.
Computational and numerical experiments show that ξ‚€πœ‹24log2ξ‚βˆ’πœ‹2=0.968β‹―>0.(2.4)
Inequalities (2.3) and (2.4) imply that 𝑓(𝑠) is strictly increasing in [1,+∞). Therefore, Lemma 2.1 follows from (2.1) and (2.2) together with the monotonicity of 𝑓(𝑠).

Lemma 2.2. Let 0≀𝑒≀1, 𝑠β‰₯1 and 𝑓𝑒,𝑠𝑠(π‘₯)=2ξ€·log1βˆ’π‘’π‘₯2ξ€Έβˆ’logπ‘₯+log(arcsinπ‘₯).(2.5)
Then inequality 𝑓𝑒,𝑠(π‘₯)>0 holds for all π‘₯∈(0,1) if and only if 3𝑠𝑒≀1, and inequality 𝑓𝑒,𝑠(π‘₯)<0 holds for all π‘₯∈(0,1) if and only if 𝑒+(2/πœ‹)2/𝑠β‰₯1.

Proof. If 𝑒=0, then we clearly see that 3𝑠𝑒≀1, 𝑒+(2/πœ‹)2/𝑠<1 and 𝑓0,𝑠(π‘₯)=log[(arcsinπ‘₯)/π‘₯]>0 for all 𝑠β‰₯1 and π‘₯∈(0,1). In the following discussion, we assume that 0<𝑒≀1.
From (2.5) and simple computations we have limπ‘₯β†’0+𝑓𝑒,𝑠𝑓(π‘₯)=0,(2.6)ξ…žπ‘’,𝑠1(π‘₯)=√1βˆ’π‘₯2βˆ’arcsinπ‘₯1+𝑒(π‘ βˆ’1)π‘₯2π‘₯ξ€·1βˆ’π‘’π‘₯2ξ€Έ=1+𝑒(π‘ βˆ’1)π‘₯2π‘₯ξ€·1βˆ’π‘’π‘₯2𝑔arcsinπ‘₯𝑒,𝑠(π‘₯),(2.7) where 𝑔𝑒,𝑠π‘₯ξ€·(π‘₯)=1βˆ’π‘’π‘₯2ξ€Έβˆš1βˆ’π‘₯2ξ€Ί1+𝑒(π‘ βˆ’1)π‘₯2ξ€»π‘”βˆ’arcsinπ‘₯,(2.8)𝑒,𝑠𝑔(0)=0,(2.9)ξ…žπ‘’,𝑠π‘₯(π‘₯)=2ξ€·1βˆ’π‘₯2ξ€Έ3/2ξ€Ί1+𝑒(π‘ βˆ’1)π‘₯2ξ€»2β„Žπ‘’,𝑠(π‘₯),(2.10) where β„Žπ‘’,𝑠(π‘₯)=𝑒2(π‘ βˆ’1)2π‘₯4ξ€·+π‘’βˆ’π‘ 2ξ€Έπ‘₯𝑒+𝑒𝑠+4π‘ βˆ’22β„Ž+1βˆ’3𝑠𝑒,(2.11)𝑒,π‘ β„Ž(0)=1βˆ’3𝑠𝑒,(2.12)𝑒,𝑠(1)=𝑒𝑠(1βˆ’π‘’)+(1βˆ’π‘’)2.(2.13)
We divide the proof into four cases.
Case 1 (3𝑠𝑒≀1). Then from (2.11) and (2.12) together with the fact that βˆ’π‘’π‘ 2+𝑒𝑠+4π‘ βˆ’2=2(π‘ βˆ’1)+𝑠(𝑒+2𝑠𝑒+1)+𝑠(1βˆ’3𝑠𝑒)>0,(2.14) we clearly see that β„Žπ‘’,𝑠(0)β‰₯0,(2.15) and β„Žπ‘’,𝑠(π‘₯) is strictly increasing in [0,1].
Equation (2.12) and the monotonicity of β„Žπ‘’,𝑠(π‘₯) imply that β„Žπ‘’,𝑠(π‘₯)>0(2.16) for π‘₯∈(0,1].
Equation (2.10) and inequality (2.16) lead to the conclusion that 𝑔𝑒,𝑠(π‘₯) is strictly increasing in [0,1). Then from (2.9) we know that 𝑔𝑒,𝑠(π‘₯)>0(2.17) for π‘₯∈(0,1).
It follows from (2.7) and inequality (2.17) that 𝑓𝑒,𝑠(π‘₯) is strictly increasing in (0,1].
Therefore, 𝑓𝑒,𝑠(π‘₯)>0 for all π‘₯∈(0,1) follows from (2.6) and the monotonicity of 𝑓𝑒,𝑠(π‘₯).
Case 2 (3𝑠𝑒>1). Then (2.12) and the continuity of β„Žπ‘’,𝑠(π‘₯) imply that there exists 0<πœ†<1 such that β„Žπ‘’,𝑠(π‘₯)<0(2.18) for π‘₯∈[0,πœ†).
Therefore, 𝑓𝑒,𝑠(π‘₯)<0 for π‘₯∈(0,πœ†) follows easily from (2.6), (2.7), (2.9) and (2.10) together with inequality (2.18).
Case 3 (𝑒+(2/πœ‹)2/𝑠β‰₯1). Then Lemma 2.1 and (2.12) lead to β„Žπ‘’,𝑠2(0)=1βˆ’3𝑠𝑒≀1βˆ’3𝑠1βˆ’πœ‹ξ‚2/𝑠<0.(2.19)
We divide the proof into two subcases.
Subcase 3.1 (𝑒=1). Then (2.13) becomes β„Žπ‘’,𝑠(1)=0.(2.20)
Let 𝑑=π‘₯2, then from (2.11) we clearly see that the function β„Žπ‘’,𝑠 is a quadratic function of variable 𝑑. It follows from inequality (2.19) and (2.20) that β„Žπ‘’,𝑠(π‘₯)<0(2.21) for all π‘₯∈[0,1).
Therefore, 𝑓𝑒,𝑠(π‘₯)<0 for π‘₯∈(0,1) follows easily from (2.6), (2.7), (2.9) and (2.10) together with inequality (2.21).
Subcase 3.2 (0<𝑒<1). Then from (2.5), (2.8), and (2.13) we have 𝑓𝑒,π‘ ξ‚ƒπœ‹(1)=log2(1βˆ’π‘’)𝑠/2≀0,(2.22)limπ‘₯β†’1βˆ’π‘”π‘’,𝑠(β„Žπ‘₯)=+∞,(2.23)𝑒,𝑠(1)>0.(2.24)
From (2.11), (2.19), and (2.24) we clearly see that there exists 0<πœ†1<1 such that β„Žπ‘’,𝑠(π‘₯)<0 for π‘₯∈[0,πœ†1) and β„Žπ‘’,𝑠(π‘₯)>0 for π‘₯∈(πœ†1,1]. Then (2.10) implies that 𝑔𝑒,𝑠(π‘₯) is strictly decreasing in [0,πœ†1] and strictly increasing in [πœ†1,1).
From (2.9) and (2.23) together with the piecewise monotonicity of 𝑔𝑒,𝑠(π‘₯) we clearly see that there exists 0<πœ†2<1 such that 𝑔𝑒,𝑠(π‘₯)<0, for π‘₯∈(0,πœ†2) and 𝑔𝑒,𝑠(π‘₯)>0 for π‘₯∈(πœ†2,1). Then (2.7) implies that 𝑓𝑒,𝑠(π‘₯) is strictly decreasing in (0,πœ†2] and strictly increasing in [πœ†2,1].
Therefore, 𝑓𝑒,𝑠(π‘₯)<0 for π‘₯∈(0,1) follows from (2.6) and (2.22) together with the piecewise monotonicity of 𝑓𝑒,𝑠(π‘₯).
Case 4 (𝑒+(2/πœ‹)2/𝑠<1). Then (2.5) leads to 𝑓𝑒,π‘ ξ‚ƒπœ‹(1)=log2(1βˆ’π‘’)𝑠/2ξ‚„>0.(2.25)
From inequality (2.25) and the continuity of 𝑓𝑒,𝑠(π‘₯) we know that there exists 0<πœ‡<1 such that 𝑓𝑒,𝑠(π‘₯)>0 for π‘₯∈(πœ‡,1].

Theorem 2.3. If 𝑑1,𝑑2∈(0,1/2) and 𝑠β‰₯1, then the double inequality 𝑄𝑑1,𝑠(π‘Ž,𝑏)<𝑃(π‘Ž,𝑏)<𝑄𝑑2,𝑠(π‘Ž,𝑏)(2.26) holds for all π‘Ž,𝑏>0 with π‘Žβ‰ π‘ if and only if 𝑑1βˆšβ‰€(1βˆ’1βˆ’(2/πœ‹)2/𝑠)/2 and 𝑑2√β‰₯(1βˆ’1/3𝑠)/2.

Proof. Since both 𝑄𝑑,𝑠(π‘Ž,𝑏) and 𝑃(π‘Ž,𝑏) are symmetric and homogeneous of degree 1. Without loss of generality, we assume that π‘Ž>𝑏. Let π‘₯=(π‘Žβˆ’π‘)/(π‘Ž+𝑏)∈(0,1). Then from (1.2) and (1.4) we have 𝑄log𝑑,𝑠(π‘Ž,𝑏)𝑄𝑃(π‘Ž,𝑏)=log𝑑,𝑠(π‘Ž,𝑏)𝐴(π‘Ž,𝑏)βˆ’log𝑃(π‘Ž,𝑏)ξ‚Ά=𝑠𝐴(π‘Ž,𝑏)2ξ€Ίlog1βˆ’(1βˆ’2𝑑)2π‘₯2ξ€»βˆ’logπ‘₯+log(arcsinπ‘₯).(2.27)
Therefore, Theorem 2.3 follows easily from Lemma 2.2 and (2.27).

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grant 11071069 and the Natural Science Foundation of Hunan Province under Grant 09JJ6003.

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