Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 605259 | https://doi.org/10.1155/2011/605259

Yu-Ming Chu, Miao-Kun Wang, Song-Liang Qiu, Ye-Fang Qiu, "Sharp Generalized Seiffert Mean Bounds for Toader Mean", Abstract and Applied Analysis, vol. 2011, Article ID 605259, 8 pages, 2011. https://doi.org/10.1155/2011/605259

Sharp Generalized Seiffert Mean Bounds for Toader Mean

Academic Editor: Detlev Buchholz
Received04 Jun 2011
Revised10 Aug 2011
Accepted11 Aug 2011
Published13 Oct 2011

Abstract

For 𝑝∈[0,1], the generalized Seiffert mean of two positive numbers ğ‘Ž and 𝑏 is defined by 𝑆𝑝(ğ‘Ž,𝑏)=𝑝(ğ‘Žâˆ’ğ‘)/arctan[2𝑝(ğ‘Žâˆ’ğ‘)/(ğ‘Ž+𝑏)],0<𝑝≤1,ğ‘Žâ‰ ğ‘;(ğ‘Ž+𝑏)/2,𝑝=0,ğ‘Žâ‰ ğ‘;ğ‘Ž,ğ‘Ž=𝑏. In this paper, we find the greatest value 𝛼 and least value 𝛽 such that the double inequality 𝑆𝛼(ğ‘Ž,𝑏)<𝑇(ğ‘Ž,𝑏)<𝑆𝛽(ğ‘Ž,𝑏) holds for all ğ‘Ž,𝑏>0 with ğ‘Žâ‰ ğ‘, and give new bounds for the complete elliptic integrals of the second kind. Here, ∫𝑇(ğ‘Ž,𝑏)=(2/𝜋)0𝜋/2âˆšğ‘Ž2cos2𝜃+𝑏2sin2𝜃𝑑𝜃 denotes the Toader mean of two positive numbers ğ‘Ž and 𝑏.

1. Introduction

For 𝑝∈[0,1], the generalized Seiffert mean of two positive numbers ğ‘Ž and 𝑏 is defined by ğ‘†ğ‘âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©(ğ‘Ž,𝑏)=𝑝(ğ‘Žâˆ’ğ‘)[]arctan2𝑝(ğ‘Žâˆ’ğ‘)/(ğ‘Ž+𝑏),0<𝑝≤1,ğ‘Žâ‰ ğ‘,ğ‘Ž+𝑏2,𝑝=0,ğ‘Žâ‰ ğ‘,ğ‘Ž,ğ‘Ž=𝑏.(1.1)

It is well known that 𝑆𝑝(ğ‘Ž,𝑏) is continuous and strictly increasing with respect to 𝑝∈[0,1] for fixed ğ‘Ž,𝑏>0 with ğ‘Žâ‰ ğ‘. In particular, if 𝑝=1/2, then the generalized Seiffert mean reduces to the Seiffert mean 𝑆(ğ‘Ž,𝑏)=ğ‘Žâˆ’ğ‘2arctan((ğ‘Žâˆ’ğ‘)/(ğ‘Ž+𝑏)),ğ‘Žâ‰ ğ‘,ğ‘Ž,ğ‘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(ğ‘Ž,𝑏) of two positive numbers ğ‘Ž and 𝑏 as follows: 2𝑇(ğ‘Ž,𝑏)=𝜋0𝜋/2âˆšğ‘Ž2cos2𝜃+𝑏2sin2=âŽ§âŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽ©î‚€âˆšğœƒğ‘‘ğœƒ,2ğ‘Žâ„°1−(𝑏/ğ‘Ž)2𝜋√,ğ‘Ž>𝑏,2𝑏ℰ1−(ğ‘Ž/𝑏)2𝜋,ğ‘Ž<𝑏,ğ‘Ž,ğ‘Ž=𝑏,(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 𝑀3/2(ğ‘Ž,𝑏)<𝑇(ğ‘Ž,𝑏)(1.4) for all ğ‘Ž,𝑏>0 with ğ‘Žâ‰ ğ‘, where ğ‘€ğ‘âŽ§âŽªâŽ¨âŽªâŽ©î‚µğ‘Ž(ğ‘Ž,𝑏)=𝑝+𝑏𝑝21/𝑝√,𝑝≠0,ğ‘Žğ‘,𝑝=0(1.5) is the power mean of order 𝑝 of two positive numbers ğ‘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(ğ‘Ž,𝑏)<𝑀log2/log(𝜋/2)(ğ‘Ž,𝑏)(1.6) for all ğ‘Ž,𝑏>0 with ğ‘Žâ‰ ğ‘.

The main purpose of this paper is to find the greatest value 𝛼 and least value 𝛽 such that the double inequality 𝑆𝛼(ğ‘Ž,𝑏)<𝑇(ğ‘Ž,𝑏)<𝑆𝛽(ğ‘Ž,𝑏) holds for all ğ‘Ž,𝑏>0 with ğ‘Žâ‰ ğ‘ 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−1/2𝜋𝑑𝑡,𝒦(0)=2,𝒦(1−ℰ)=+∞,(𝑟)=0𝜋/21−𝑟2sin2𝑡1/2𝑑𝑡,ℰ(0)=𝜋/2,ℰ(1−)=1,𝑑𝒦(𝑟)=𝑑𝑟ℰ(𝑟)−1−𝑟2𝒦(𝑟)𝑟1−𝑟2,𝑑ℰ(𝑟)=ℰ𝑑𝑟(𝑟)−𝒦(𝑟)𝑟,𝑑ℰ(𝑟)−1−𝑟2𝒦(𝑟)𝑑[]𝑑𝑟=𝑟𝒦(𝑟),𝒦(𝑟)−ℰ(𝑟)=𝑑𝑟𝑟ℰ(𝑟)1−𝑟2,ℰ2√𝑟=1+𝑟2ℰ(𝑟)−1−𝑟2𝒦(𝑟).1+𝑟(2.1)

Lemma 2.1 (see [10, Theorem 1.25]). For −∞<ğ‘Ž<𝑏<∞, let 𝑓(𝑥),𝑔(𝑥)∶[ğ‘Ž,𝑏]→ℝ be continuous on [ğ‘Ž,𝑏] and be differentiable on (ğ‘Ž,𝑏), let ğ‘”î…ž(𝑥)≠0 for all 𝑥∈(ğ‘Ž,𝑏). If ğ‘“î…ž(𝑥)/ğ‘”î…ž(𝑥) is increasing (decreasing) on (ğ‘Ž,𝑏), then so are 𝑓(𝑥)−𝑓(ğ‘Ž),𝑔(𝑥)−𝑔(ğ‘Ž)𝑓(𝑥)−𝑓(𝑏)𝑔(𝑥)−𝑔(𝑏).(2.2) If ğ‘“î…ž(𝑥)/ğ‘”î…ž(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 ğ¹î…ž1(𝑟)ğ¹î…ž2=(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 ğºî…ž1(𝑟)2ğºî…ž2=(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 ğ‘”î…ž2√(𝑟)=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 𝑔18(𝑟)=ℰ⋅ℰ(𝑟)(𝑟)−1−𝑟2𝒦(𝑟)𝑟22+ℰ16(𝑟)−1−𝑟2𝒦(𝑟)/𝑟2−𝜋/4𝑟2−4𝜋−𝜋𝑟2−8ℰ(𝑟)𝑟4>16𝜋⋅𝜋42𝜋+16⋅32−(3𝜋−8)=8−3𝜋2>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 lim𝑟→0+𝑓(𝑟)=lim𝑟→1−𝑓(𝑟)=0,(2.7)𝑓𝑓′(𝑟)=1(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 lim𝑟→0+𝑓1(𝑟)=0,(2.9)lim𝑟→1−𝑓1𝑓(𝑟)=8−2𝜋>0,(2.10)14(𝑟)=2ℰ(𝑟)−1−𝑟2𝒦(𝑟)ℰ(𝑟)−1−𝑟2𝒦(𝑟)𝑟−2𝜋𝑟ℰ(𝑟)−𝜋1+𝑟2ℰ(𝑟)−𝒦(𝑟)𝑟=𝑟𝑓2(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, lim𝑟→0+𝑓2𝜋(𝑟)=−24,(2.12)lim𝑟→1−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(ğ‘Ž,𝑏)<𝑇(ğ‘Ž,𝑏)<𝑆1/2(ğ‘Ž,𝑏) holds for all ğ‘Ž,𝑏>0 with ğ‘Žâ‰ ğ‘, and 𝑆√3/4(ğ‘Ž,𝑏) and 𝑆1/2(ğ‘Ž,𝑏) are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean 𝑇(ğ‘Ž,𝑏), respectively.

Proof. Firstly, we prove that 𝑆√3/4(ğ‘Ž,𝑏)<𝑇(ğ‘Ž,𝑏)<𝑆1/2(ğ‘Ž,𝑏)(3.1) for all ğ‘Ž,𝑏>0 with ğ‘Žâ‰ ğ‘.
Without loss of generality, we assume that ğ‘Ž>𝑏. Let 𝑡=𝑏/ğ‘Ž<1, 𝑟=(1−𝑡)/(1+𝑡). Then (1.1) and (1.3) lead to 𝑇(ğ‘Ž,𝑏)−𝑆√3/4(ğ‘Ž,𝑏)=2ğ‘Žğœ‹â„°î‚€âˆš1−𝑡2−√3ğ‘Ž(1−𝑡)√4arctan3=(1−𝑡)/2(1+𝑡)2ğ‘Žğœ‹â„°îƒ©2√𝑟−√1+𝑟3ar√2(1+𝑟)arctan𝑟=3/22ğ‘Žğœ‹î€ºî€·2ℰ(𝑟)−1−𝑟2𝒦(𝑟)−√1+𝑟3ar√2(1+𝑟)arctan𝑟=3/22ğ‘Ž2ℰ(𝑟)−1−𝑟2𝒦(𝑟)√𝜋(1+𝑟)arctan𝑟3/2𝑔(𝑟),(3.2)𝑇(ğ‘Ž,𝑏)−𝑆1/2(ğ‘Ž,𝑏)=2ğ‘Žğœ‹â„°î‚€âˆš1−𝑡2î‚âˆ’ğ‘Ž(1−𝑡)=2arctan((1−𝑡)/(1+𝑡))2ğ‘Žğœ‹â„°îƒ©2√𝑟−1+ğ‘Ÿğ‘Žğ‘Ÿ=(1+𝑟)arctan𝑟2ğ‘Žğœ‹î€ºî€·2ℰ(𝑟)−1−𝑟2𝒦(𝑟)−1+𝑟ar=(1+𝑟)arctan𝑟2ğ‘Ž2ℰ(𝑟)−1−𝑟2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(ğ‘Ž,𝑏) and 𝑆1/2(ğ‘Ž,𝑏) are the best possible lower and upper generalized Seiffert mean bounds for the Toader mean 𝑇(ğ‘Ž,𝑏), respectively.
For any 𝜀>0 and 0<𝑥<1, from (1.1) and (1.3) one has lim𝑥→0𝑆1/2−𝜀=(1,𝑥)−𝑇(1,𝑥)1−2𝜀−22arctan(1−2𝜀)𝜋<1−22arctan1𝜋𝑆=0,(3.4)√3/4+𝜀(1,1−𝑥)−𝑇(1,1−𝑥)=𝐽(𝑥)√arctan,3+4𝜀𝑥/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√+𝜀𝑥−34𝑥1+𝜀1−21𝑥+𝑥162𝑥+𝑜2×⎧⎪⎨⎪⎩11+2⎡⎢⎢⎣1𝑥+4−13√34+𝜀2âŽ¤âŽ¥âŽ¥âŽ¦ğ‘¥2𝑥+𝑜2⎫⎪⎬⎪⎭=𝜀3√32√+𝜀34𝑥+𝜀3𝑥+𝑜3.(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 𝑆1/2−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 √√3𝜋1−1−𝑟2√8arctan3√1−1−𝑟2/2√1+1−𝑟2𝜋√<𝐸(𝑟)<1−1−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.

References

  1. H. J. Seiffert, “Aufgabe β16,” Die Wurzel, vol. 29, pp. 221–222, 1995. View at: Google Scholar
  2. G. Toader, “Seiffert type means,” Nieuw Archief voor Wiskunde, vol. 17, no. 3, pp. 379–382, 1999. View at: Google Scholar
  3. P. A. Hästö, “A monotonicity property of ratios of symmetric homogeneous means,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 5, article 71, p. 23, 2002. View at: Google Scholar | Zentralblatt MATH
  4. Y.-M. Chu, M.-K. Wang, and Y.-F. Qiu, “An optimal double inequality between power-type Heron and Seiffert means,” Journal of Inequalities and Applications, vol. 2010, Article ID 146945, 11 pages, 2010. View at: Google Scholar | Zentralblatt MATH
  5. J. Sándor, “On certain inequalities for means. III,” Archiv der Mathematik, vol. 76, no. 1, pp. 34–40, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. G. Toader, “Some mean values related to the arithmetic-geometric mean,” Journal of Mathematical Analysis and Applications, vol. 218, no. 2, pp. 358–368, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. M. Vuorinen, “Hypergeometric functions in geometric function theory,” in Special Functions and Differential Equations (Madras, 1997), pp. 119–126, Allied, New Delhi, India, 1998. View at: Google Scholar | Zentralblatt MATH
  8. R. W. Barnard, K. Pearce, and K. C. Richards, “A monotonicity property involving 3F2 and comparisons of the classical approximations of elliptical arc length,” SIAM Journal on Mathematical Analysis, vol. 32, no. 2, pp. 403–419, 2000. View at: Publisher Site | Google Scholar
  9. H. Alzer and S.-L. Qiu, “Monotonicity theorems and inequalities for the complete elliptic integrals,” Journal of Computational and Applied Mathematics, vol. 172, no. 2, pp. 289–312, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. G. D. Anderson, M. K. Vamanamurthy, and M. K. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, NY, USA, 1997.

Copyright © 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views819
Downloads515
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.