Sharp Generalized Seiffert Mean Bounds for Toader Mean

Abstract

For 𝑝∈[0,1], the generalized Seiffert mean of two positive numbers 𝑎 and 𝑏 is defined by 𝑆_𝑝(𝑎,𝑏)=𝑝(𝑎−𝑏)/arctan[2𝑝(𝑎−𝑏)/(𝑎+𝑏)],0<𝑝≤1,𝑎≠𝑏;(𝑎+𝑏)/2,𝑝=0,𝑎≠𝑏;𝑎,𝑎=𝑏. In this paper, we find the greatest value 𝛼 and least value 𝛽 such that the double inequality 𝑆_𝛼(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝑆_𝛽(𝑎,𝑏) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏, and give new bounds for the complete elliptic integrals of the second kind. Here, 𝑇(𝑎,𝑏)=(2/𝜋)∫^0𝜋/2√𝑎²cos²𝜃+𝑏²sin²𝜃𝑑𝜃 denotes the Toader mean of two positive numbers 𝑎 and 𝑏.

1. Introduction

For 𝑝∈[0,1], the generalized Seiffert mean of two positive numbers 𝑎 and 𝑏 is defined by 𝑆_𝑝(𝑎,𝑏)=⎧⎪⎪⎨⎪⎪⎩𝑝(𝑎−𝑏)a𝑎r+cta𝑏n[2𝑝(𝑎−𝑏)/(𝑎+𝑏)],0<𝑝≤1,𝑎≠𝑏,𝑎,2,𝑝𝑎==0𝑏.,𝑎≠𝑏,(1.1)

It is well known that 𝑆_𝑝(𝑎,𝑏) is continuous and strictly increasing with respect to 𝑝∈[0,1] for fixed 𝑎,𝑏>0 with 𝑎≠𝑏. In particular, if 𝑝=1/2, then the generalized Seiffert mean reduces to the Seiffert mean 𝑆(𝑎,𝑏)=𝑎2,arctan((𝑎𝑎−−𝑏𝑏)/(𝑎+𝑏)),𝑎𝑎≠=𝑏𝑏,.(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(𝑎,𝑏) of two positive numbers 𝑎 and 𝑏 as follows: 𝑇(𝑎,𝑏)=2𝜋^0𝜋/2=⎧⎪⎪⎪⎨⎪⎪⎪⎩2𝑎ℰ√√𝑎²cos²𝜃+𝑏²sin²𝜃𝑑𝜃,1−(𝑏/𝑎)²2𝑏ℰ√𝜋,𝑎>𝑏,1−(𝑎/𝑏)²𝑎,𝜋,𝑎𝑎<=𝑏𝑏,,(1.3) where ℰ(𝑟)=∫^0𝜋/2(1−𝑟²sin²𝑡)^1/2𝑑𝑡, 𝑟∈[0,1) is the complete elliptic integral of the second kind.

Vuorinen [7] conjectured that 𝑀_3/2(𝑎,𝑏)<𝑇(𝑎,𝑏)(1.4) for all 𝑎,𝑏>0 with 𝑎≠𝑏, where 𝑀_𝑝(𝑎,𝑏)=⎧⎪⎨⎪⎩𝑎^𝑝+𝑏^𝑝2^1/𝑝√,𝑝≠0,𝑎𝑏,𝑝=0(1.5) is the power mean of order 𝑝 of two positive numbers 𝑎 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(𝑎,𝑏)<𝑀_{log2/log(𝜋/2)}(𝑎,𝑏)(1.6) for all 𝑎,𝑏>0 with 𝑎≠𝑏.

The main purpose of this paper is to find the greatest value 𝛼 and least value 𝛽 such that the double inequality 𝑆_𝛼(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝑆_𝛽(𝑎,𝑏) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏 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 𝑟∈[0,1), then 𝒦(𝑟)=^0𝜋/21−𝑟²sin²𝑡^−1/2𝑑𝑡,𝒦(0)=𝜋2,𝒦(1⁻ℰ(𝑟)=_0𝜋/2)=+∞,𝑑𝒦(𝑟)1−𝑟²sin²𝑡^1/2𝑑𝑡,ℰ(0)=𝜋/2,ℰ(1⁻)=1,𝑑𝑟=ℰ(𝑟)−1−𝑟²𝒦(𝑟)𝑑ℰ(𝑟)−1−𝑟₂𝑟1−𝑟²,𝑑ℰ𝑑𝑟(𝑟)=ℰ(𝑟)−𝑟𝒦(𝑟),𝒦(𝑟)𝑑𝑟=𝑟𝒦(𝑟),𝑑[𝒦(𝑟𝑑)𝑟−ℰ(𝑟)]=1𝑟ℰ−(𝑟𝑟²)ℰ2√,𝑟1+𝑟=2ℰ(𝑟)−1−𝑟²𝒦(𝑟)1+𝑟.(2.1)

Lemma 2.1 (see [10, Theorem 1.25]). For −∞<𝑎<𝑏<∞, let 𝑓(𝑥),𝑔(𝑥)∶[𝑎,𝑏]→ℝ be continuous on [𝑎,𝑏] and be differentiable on (𝑎,𝑏), let 𝑔^(𝑥)≠0 for all 𝑥∈(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.

Lemma 2.2. (1)  [ℰ(𝑟)−(1−𝑟²)𝒦(𝑟)]/𝑟² is strictly increasing from (0,1) onto (𝜋/4,1);
(2)    {[ℰ(𝑟)−(1−𝑟²)𝒦(𝑟)]/𝑟²−𝜋/4}/𝑟² is strictly increasing from (0,1) onto (𝜋/32,1−𝜋/4);
(3)    [𝒦(𝑟)−ℰ(𝑟)]/𝑟² is strictly increasing from (0,1) onto (𝜋/4,+∞);
(4)    2ℰ(𝑟)−(1−𝑟²)𝒦(𝑟) is strictly increasing from (0,1) onto (𝜋/2,2);
(5)    𝐹(𝑟)=[(2−𝑟²)𝒦(𝑟)−2ℰ(𝑟)]/𝑟⁴ is strictly increasing from (0,1) onto (𝜋/16,+∞);
(6)    𝐺(𝑟)=[4𝜋−𝜋𝑟²−8ℰ(𝑟)]/𝑟⁴ is strictly increasing from (0,1) onto (3𝜋/16,3𝜋−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(1⁻)=+∞. Let 𝐹₁(𝑟)=(2−𝑟²)𝒦(𝑟)−2ℰ(𝑟) and 𝐹₂(𝑟)=𝑟⁴, then 𝐹(𝑟)=𝐹₁(𝑟)/𝐹₂(𝑟), 𝐹₁(0)=𝐹₂(0)=0 and 𝐹^1(𝑟)𝐹^2(𝑟)=ℰ(𝑟)−1−𝑟²𝒦4𝑟²1−𝑟².(2.3)
It follows from (2.3) and part (1) together with Lemma 2.1 that 𝐹(𝑟) is strictly increasing in (0,1) and 𝐹(0⁺)=𝜋/16.
For part (6), clearly 𝐺(1⁻)=3𝜋−8. Let 𝐺₁(𝑟)=4𝜋−𝜋𝑟²−8ℰ(𝑟) and 𝐺₂(𝑟)=𝑟⁴, then 𝐺(𝑟)=𝐺₁(𝑟)/𝐺₂(𝑟), 𝐺₁(0)=𝐺₂(0)=0, and 𝐺^1(𝑟)2𝐺^2(𝑟)=2−𝑟²𝒦(𝑟)−2ℰ(𝑟)𝑟⁴+ℰ(𝑟)−1−𝑟²𝒦(𝑟)/𝑟²−𝜋/4𝑟².(2.4)
From (2.4), parts (2) and (5) together with Lemma 2.1, we know that 𝐺(𝑟) is strictly increasing in (0,1), and 𝑓(0⁺)=3𝜋/16.

Lemma 2.3. (1)𝑔(𝑟)=arctan(√3𝑟/2)−√3𝜋𝑟/{4[2ℰ(𝑟)−(1−𝑟²)𝒦(𝑟)]} is strictly increasing from (0,1) onto (0,arctan(√3/2)−√3𝜋/8).
(2)𝑓(𝑟)=arctan𝑟−𝜋𝑟/{2[2ℰ(𝑟)−(1−𝑟²)𝒦(𝑟)]}<0 for 𝑟∈(0,1).

Proof. For part (1), clearly 𝑔(0⁺)=0 and 𝑔(1⁻)=arctan(√3/2)−√3𝜋/8=0.0335⋯>0. Simple computation leads to 𝑔^(𝑟)=2√34+3𝑟²−√3𝜋ℰ(𝑟)42ℰ(𝑟)−1−𝑟²𝒦(𝑟)²=√3𝑟⁴ℰ(𝑟)44+3𝑟²2ℰ(𝑟)−1−𝑟²𝒦(𝑟)²𝑔₁(𝑟),(2.5) where 𝑔₁(𝑟)={8[2ℰ(𝑟)−(1−𝑟²)𝒦(𝑟)]²−𝜋(4+3𝑟²)ℰ(𝑟)}/[𝑟⁴ℰ(𝑟)].
Making use of Lemma 2.2 (1), (2), and (6), we get 𝑔₁(𝑟)=8ℰ(𝑟)⋅ℰ(𝑟)−1−𝑟²𝒦(𝑟)𝑟²²+16ℰ(𝑟)−1−𝑟²𝒦(𝑟)/𝑟²−𝜋/4𝑟²−4𝜋−𝜋𝑟²−8ℰ(𝑟)𝑟⁴>16𝜋⋅𝜋4²+16⋅𝜋32−(3𝜋−8)=8−32𝜋>0.(2.6)
Therefore, part (1) follows from (2.5) and (2.6) together with the limiting values of 𝑔(𝑟) at 𝑟=0 and 𝑟=1.
For part (2), simple computations yield that lim_𝑟→0+𝑓(𝑟)=lim_𝑟→1−𝑓(𝑟)=0,(2.7)𝑓′(𝑟)=𝑓¹(𝑟)21+𝑟²2ℰ(𝑟)−1−𝑟²𝒦(𝑟)²,(2.8) where 𝑓₁(𝑟)=2[2ℰ(𝑟)−(1−𝑟²)𝒦(𝑟)]²−𝜋(1+𝑟²)ℰ(𝑟). Note that lim_𝑟→1−lim_𝑟→0+𝑓₁(𝑟)=0,(2.9)𝑓𝑓₁(𝑟)=8−2𝜋>0,(2.10)_1(𝑟)=42ℰ(𝑟)−1−𝑟²𝒦(𝑟)ℰ(𝑟)−1−𝑟²𝒦(𝑟)−𝜋1+𝑟₂𝑟−2𝜋𝑟ℰ(𝑟)=𝑟𝑓₂ℰ(𝑟)−𝑟𝒦(𝑟)(𝑟),(2.11) where 𝑓₂(𝑟)=4[2ℰ(𝑟)−(1−𝑟²)𝒦(𝑟)][ℰ(𝑟)−(1−𝑟²)𝒦(𝑟)]/𝑟²−2𝜋ℰ(𝑟)+𝜋(1+𝑟²)[𝒦(𝑟)−ℰ(𝑟)]/𝑟².
From Lemma 2.2(1), (3), and (4) together with the monotonicity of ℰ(𝑟) we know that 𝑓₂(𝑟) is strictly increasing in (0,1). Moreover, lim_𝑟→0+𝑓₂(𝑟)=−𝜋²lim_𝑟→1−4,(2.12)𝑓₂(𝑟)=+∞.(2.13)
Equations (2.11)–(2.13) and the monotonicity of 𝑓₂(𝑟) lead to the conclusion that there exists 𝑟₀∈(0,1) such that 𝑓₁(𝑟) is strictly decreasing in (0,𝑟₀) and strictly increasing in (𝑟₀,1).
It follows from (2.8)–(2.10) and the piecewise monotonicity of 𝑓₁(𝑟) that there exists 𝑟₁∈(0,1) such that 𝑓(𝑟) is strictly decreasing in (0,𝑟₁) and strictly increasing in (𝑟₁,1).
Therefore, part (2) follows from (2.7) and the piecewise monotonicity of 𝑓(𝑟).

3. Main Result

Theorem 3.1. Inequality 𝑆_√3/4(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝑆_1/2(𝑎,𝑏) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏, and 𝑆_√3/4(𝑎,𝑏) and 𝑆_1/2(𝑎,𝑏) are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean 𝑇(𝑎,𝑏), respectively.

Proof. Firstly, we prove that 𝑆_√3/4(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝑆_1/2(𝑎,𝑏)(3.1) for all 𝑎,𝑏>0 with 𝑎≠𝑏.
Without loss of generality, we assume that 𝑎>𝑏. Let 𝑡=𝑏/𝑎<1, 𝑟=(1−𝑡)/(1+𝑡). Then (1.1) and (1.3) lead to 𝑇(𝑎,𝑏)−𝑆_√3/4(𝑎,𝑏)=2𝑎𝜋ℰ√1−𝑡²−√3𝑎(1−𝑡)=2𝑎4arctan√3(1−𝑡)/2(1+𝑡)𝜋ℰ2√𝑟1+𝑟−√3ar=2𝑎2(1+𝑟)arctan√3/2𝑟𝜋2ℰ(𝑟)−1−𝑟²𝒦(𝑟)1+𝑟−√3ar=2𝑎2ℰ(𝑟)−1−𝑟₂2(1+𝑟)arctan√3/2𝑟𝒦(𝑟)𝑇(𝑎,𝑏)−𝑆_1/2𝜋(1+𝑟)arctan√3/2𝑟𝑔(𝑟),(3.2)(𝑎,𝑏)=2𝑎𝜋ℰ√1−𝑡²−𝑎(1−𝑡)=2𝑎2arctan((1−𝑡)/(1+𝑡))𝜋ℰ2√𝑟=2𝑎1+𝑟−(1+𝑟)𝑎a𝑟rctan𝑟𝜋2ℰ(𝑟)−1−𝑟²𝒦(𝑟)1+𝑟−ar=2𝑎2ℰ(𝑟)−1−𝑟₂(1+𝑟)arctan𝑟𝒦(𝑟)𝜋(1+𝑟)arctan𝑟𝑓(𝑟),(3.3) where 𝑔(𝑟) and 𝑓(𝑟) 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 𝑆_√3/4(𝑎,𝑏) and 𝑆_1/2(𝑎,𝑏) are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean 𝑇(𝑎,𝑏), respectively.
For any 𝜀>0 and 0<𝑥<1, from (1.1) and (1.3) one has lim_𝑥→0𝑆_1/2−𝜀(1,𝑥)−𝑇(1,𝑥)=1−2𝜀𝑆2arctan(1−2𝜀)−𝜋2<2arc1tan1−𝜋2=0,(3.4)_√3/4+𝜀(1,1−𝑥)−𝑇(1,1−𝑥)=𝐽(𝑥)arctan√3+4𝜀𝑥/2(2−𝑥),(3.5) where 𝐽(𝑥)=(√3/4+𝜀)𝑥−2ℰ(√2𝑥−𝑥²)arctan{[(√3+4𝜀)𝑥]/[2(2−𝑥)]}/𝜋.
Letting 𝑥→0 and making use of Taylor expansion, we get 𝐽(𝑥)=√34+𝜀𝑥−√34+𝜀𝑥1−21𝑥+116𝑥²×⎧⎪⎨⎪⎩1+1+𝑜𝑥²2𝑥+⎡⎢⎢⎣41−13√34+𝜀²⎤⎥⎥⎦𝑥₂=𝜀+𝑜𝑥²⎫⎪⎬⎪⎭3√32+𝜀√34+𝜀𝑥³+𝑜𝑥³.(3.6)
Inequality (3.4) and equations (3.5) and (3.6) imply that for any 𝜀>0 there exist 𝛿₁=𝛿₁(𝜀)>0 and 𝛿₂=𝛿₂(𝜀)>0, such that 𝑆_√3/4+𝜀(1,1−𝑥)>𝑇(1,1−𝑥) for 𝑥∈(0,𝛿₁) and 𝑆_1/2−𝜀(1,𝑥)<𝑇(1,𝑥) for 𝑥∈(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 √3𝜋1−√1−𝑟²8arc<tan𝐸(𝑟√)<31−√1−𝑟²𝜋/12−1√+√1−𝑟²1−𝑟²4arctan1−√1−𝑟²/1+√1−𝑟²(3.7) holds for all 𝑟∈(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.