Optimal Bounds for Seiffert Mean in terms of One-Parameter Means

Abstract

The authors present the greatest value 𝑟₁ and the least value 𝑟₂ such that the double inequality 𝐽_𝑟1(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝐽_𝑟2(𝑎,𝑏) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏, where 𝑇(𝑎,𝑏) and 𝐽_𝑝(𝑎,𝑏) denote the Seiffert and pth one-parameter means of two positive numbers a and b, respectively.

1. Introduction

For 𝑝∈ℝ the 𝑝th one-parameter mean 𝐽_𝑝(𝑎,𝑏) and the Seiffert mean 𝑇(𝑎,𝑏) of two positive real numbers 𝑎 and 𝑏 are defined by 𝐽_𝑝()=𝑝𝑎𝑎,𝑏^𝑝+1−𝑏^𝑝+1𝑝+1𝑎^𝑝−𝑏^𝑝,𝑎≠𝑏,𝑝≠0,−1,𝑎−𝑏log𝑎−log𝑏,𝑎≠𝑏,𝑝=0,𝑎𝑏log𝑎−log𝑏𝑇()=𝑎−𝑏,𝑎≠𝑏,𝑝=−1,𝑎,𝑎=𝑏,𝑎,𝑏𝑎−𝑏2arctan((𝑎−𝑏)/(𝑎+𝑏)),𝑎≠𝑏,𝑎,𝑎=𝑏, (1.1)(1.2) respectively. Recently, both mean values have been the subject of intensive research. In particular, many remarkable inequalities and properties for 𝐽_𝑝 and 𝑇 can be found in the literature [1–14].

It is well known that the one-parameter mean 𝐽_𝑝(𝑎,𝑏) is continuous and strictly increasing with respect to 𝑝∈ℝ for fixed 𝑎, 𝑏>0 with 𝑎≠𝑏. Many mean values are the special case of the one-parameter mean, for example: 𝐽₁()=()𝑎,𝑏𝑎+𝑏2𝐽,thearithmeticmean,_1/2()=√𝑎,𝑏𝑎+𝑎𝑏+𝑏3𝐽,theHeronianmean,_−1/2(𝑎,𝑏)=√𝐽𝑎𝑏,thegeometricmean,₋₂(𝑎,𝑏)=2𝑎𝑏(𝑎+𝑏),theharmonicmean. (1.3)

Seiffert [4] proved that the double inequality 𝑀₁(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝑀₂(𝑎,𝑏) (1.4) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏, where 𝑀_𝑟(𝑎,𝑏)=[(𝑎^𝑟+𝑏^𝑟)/2]^1/𝑟(𝑟≠0) and 𝑀₀(𝑎,𝑏)=√𝑎𝑏 is the 𝑟th power mean of 𝑎 and 𝑏.

In [15–17], the authors presented the best possible bounds for the Seiffert mean in terms of the Lehmer, power-type Heron, and one-parameter Gini means as follows:𝐿₀(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝐿_1/3(),𝐻𝑎,𝑏_{log3/log(𝜋/2)}(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝐻_5/2(),𝑆𝑎,𝑏₁(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝑆_5/3(𝑎,𝑏). (1.5) for all 𝑎,𝑏>0 with 𝑎≠𝑏, where 𝐿_𝑟(𝑎,𝑏)=(𝑎^𝑟+1+𝑏^𝑟+1)/(𝑎^𝑟+𝑏^𝑟), 𝐻_𝑘(𝑎,𝑏)=[(𝑎^𝑘+(𝑎𝑏)^𝑘/2+𝑏^𝑘)/3]^1/𝑘(𝑘≠0) and 𝐻₀(𝑎,𝑏)=√𝑎𝑏, and 𝑆_𝛼(𝑎,𝑏)=[(𝑎^𝛼−1+𝑏^𝛼−1)/(𝑎+𝑏)]^1/(𝛼−2)(𝛼≠2) and 𝑆₂(𝑎,𝑏)=(𝑎^𝑎𝑏^𝑏)^{1/(𝑎+𝑏)} denote the Lehmer, power-type Heron, and one-parameter Gini means of 𝑎 and 𝑏, respectively.

The purpose of this paper is to answer the question: what are the greatest value 𝑟₁ and the least value 𝑟₂ such that the double inequality 𝐽_𝑟1(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝐽_𝑟2(𝑎,𝑏) (1.6) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏?

2. Lemma

In order to establish our main result we need the following lemma.

Lemma 2.1. If 𝑝=2/(𝜋−2)=1.75⋯, 𝑡≥1 and 𝑔(𝑡)=−(𝑝−1)𝑡^2𝑝+2 + (𝑝+1)𝑡^2𝑝+𝑝(𝑝+1)𝑡^𝑝+3−2(𝑝+1)²𝑡^𝑝+2 + 2𝑝(𝑝+3)𝑡^𝑝+1−2(𝑝+1)²𝑡^𝑝 + 𝑝(𝑝+1)𝑡^𝑝−1+(𝑝+1)𝑡²−(𝑝−1), then there exists 𝜆∈(1,∞) such that 𝑔(𝑡)>0 for 𝑡∈(1,𝜆) and 𝑔(𝑡)<0 for 𝑡∈(𝜆,∞).

Proof. Let 𝑔₁(𝑡)=𝑔^′(𝑡)/𝑡, 𝑔₂(𝑡)=𝑡^4−𝑝𝑔^′₁(𝑡) and 𝑔₃(𝑡)=𝑡^4−𝑝𝑔₂⁽⁵⁾(𝑡)/[4𝑝²(𝑝−1)²(𝑝+1)²]. Then simple computations lead to 𝑔(1)=0,lim_𝑡→+∞𝑔(𝑡)𝑔=−∞,₁(𝑡)=−2𝑝−1𝑝+1𝑡^2𝑝+2𝑝𝑝+1𝑡^2𝑝−2+𝑝𝑝+1𝑝+3×𝑡^𝑝+1−2𝑝+1²𝑝+2𝑡^𝑝+2𝑝𝑝+1𝑝+3𝑡^𝑝−1−2𝑝𝑝+1²𝑡^𝑝−2+𝑝𝑝+1𝑝−1𝑡^𝑝−3,𝑔+2𝑝+1₁(1)=0,lim_𝑡→+∞𝑔₁(𝑡)𝑔=−∞,₂(𝑡)=−4𝑝𝑝−1𝑝+1𝑡^𝑝+3+4𝑝𝑝+1𝑝−1𝑡^𝑝+1+𝑝𝑝+1²×𝑝+3𝑡⁴−2𝑝𝑝+1²𝑝+2𝑡³+2𝑝𝑝−1𝑝+1𝑝+3×𝑡²−2𝑝𝑝+1²,𝑔𝑝−2𝑡+𝑝𝑝+1𝑝−1𝑝−3₂(1)=0,lim_𝑡→+∞𝑔₂(𝑡)𝑔=−∞,_′2(𝑡)=−4𝑝𝑝−1𝑝+1𝑝+3𝑡^𝑝+2+4𝑝𝑝+1²𝑝−1𝑡^𝑝+4𝑝𝑝+1²𝑝+3𝑡³−6𝑝𝑝+1²𝑝+2𝑡²+4𝑝𝑝−1𝑝+1𝑝+3𝑡−2𝑝𝑝+1²,𝑔𝑝−2_′2(1)=0,lim_𝑡→+∞𝑔^′₂(𝑡)𝑔=−∞,_2′′(𝑡)=−4𝑝𝑝−1𝑝+1𝑝+2𝑝+3𝑡^𝑝+1+4𝑝²𝑝+1²×𝑝−1𝑡^𝑝−1+12𝑝𝑝+1²𝑝+3𝑡²−12𝑝𝑝+1²×,𝑔𝑝+2𝑡+4𝑝𝑝−1𝑝+1𝑝+3_2′′(1)=12𝑝2−𝑝𝑝+1²>0,lim_𝑡→+∞𝑔₂^′′(𝑡)𝑔=−∞,_2′′′(𝑡)=−4𝑝𝑝−1𝑝+1²𝑝+2𝑝+3𝑡^𝑝+4𝑝²𝑝+1²×𝑝−1²𝑡^𝑝−2+24𝑝𝑝+1²𝑝+3𝑡−12𝑝𝑝+1²,𝑔𝑝+2_2′′′(1)=12𝑝2−𝑝2𝑝+3𝑝+1²>0,lim_𝑡→+∞𝑔₂^′′′(𝑡)𝑔=−∞,₂₍₄₎(𝑡)=−4𝑝²𝑝−1𝑝+1²𝑝+2𝑝+3𝑡^𝑝−1+4𝑝²𝑝+1²×𝑝−1²𝑝−2𝑡^𝑝−3+24𝑝𝑝+1²,𝑔𝑝+3₂₍₄₎(1)=8𝑝𝑝+1²−4𝑝³+2𝑝²+5𝑝+9>8𝑝𝑝+1²−4×1.8³+2×1.75²+5×1.75+9=4.376𝑝𝑝+1²>0,lim_𝑡→+∞𝑔₂⁽⁴⁾(𝑡)𝑔=−∞,₃(𝑡)=−𝑝+2𝑝+3𝑡²++𝑝−2𝑝−3≤−𝑝+2𝑝+3𝑝−2𝑝−3=−10𝑝<0 (2.1)(2.2)(2.3)(2.4)(2.5)(2.6)(2.7)(2.8)(2.9)(2.10)(2.11)(2.12)(2.13)(2.14)(2.15)(2.16)(2.17)(2.18)(2.19)(2.20)(2.21) for 𝑡∈[1,∞).
From the inequality (2.21) we clearly see that 𝑔₂⁽⁴⁾(𝑡) is strictly decreasing in [1,∞). Then (2.19) and (2.20) lead to the conclusion that there exists 𝜆₁>1 such that 𝑔₂⁽⁴⁾(𝑡)>0 for 𝑡∈[1,𝜆₁) and 𝑔₂⁽⁴⁾(𝑡)<0 for 𝑡∈(𝜆₁,∞). Hence, 𝑔₂^′′′(𝑡) is strictly increasing in [1,𝜆₁] and strictly decreasing in [𝜆₁,∞).
It follows from (2.16) and (2.17) together with the monotonicity of 𝑔₂^′′′(𝑡) that there exists 𝜆₂>1 such that 𝑔₂^′′′(𝑡)>0 for 𝑡∈[1,𝜆₂) and 𝑔₂^′′′(𝑡)<0 for 𝑡∈(𝜆₂,∞). Therefore, 𝑔₂^′′(𝑡) is strictly increasing in [1,𝜆₂] and strictly decreasing in [𝜆₂,∞).
From (2.13) and (2.14) together with the monotonicity of 𝑔₂^′′(𝑡) we know that there exists 𝜆₃>1 such that 𝑔₂^′′(𝑡)>0 for 𝑡∈[1,𝜆₃) and 𝑔₂^′′(𝑡)<0 for 𝑡∈(𝜆₃,∞). So, 𝑔^′₂(𝑡) is strictly increasing in [1,𝜆₃] and strictly decreasing in [𝜆₃,∞).
Equations (2.10) and (2.11) together with the monotonicity of 𝑔^′₂(𝑡) imply that there exists 𝜆₄>1 such that 𝑔^′₂(𝑡)>0 for 𝑡∈(1,𝜆₄) and 𝑔^′₂(𝑡)<0 for 𝑡∈(𝜆₄,∞). Hence, 𝑔₂(𝑡) is strictly increasing in [1,𝜆₄] and strictly decreasing in [𝜆₄,∞).
It follows from (2.7) and (2.8) together with the monotonicity of 𝑔₂(𝑡) that there exists 𝜆₅>1 such that 𝑔₂(𝑡)>0 for 𝑡∈(1,𝜆₅) and 𝑔₂(𝑡)<0 for 𝑡∈(𝜆₅,∞). Therefore, 𝑔₁(𝑡) is strictly increasing in [1,𝜆₅] and strictly decreasing in [𝜆₅,∞).
From (2.4) and (2.5) together with the monotonicity of 𝑔₁(𝑡) we clearly see that there exists 𝜆₆>1 such that 𝑔₁(𝑡)>0 for 𝑡∈(1,𝜆₆) and 𝑔₁(𝑡)<0 for 𝑡∈(𝜆₆,∞). So, 𝑔(𝑡) is strictly increasing in [1,𝜆₆] and strictly decreasing in [𝜆₆,∞).
Therefore, Lemma 2.1 follows from (2.1) and (2.2) together with the monotonicity of 𝑔(𝑡).

3. Main Result

Theorem 3.1. The double inequality 𝐽_2/(𝜋−2)(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝐽₂(𝑎,𝑏) (3.1) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏, and 𝐽_2/(𝜋−2)(𝑎,𝑏) and 𝐽₂(𝑎,𝑏) are the best possible lower and upper one-parameter mean bounds for the Seiffert mean 𝑇(𝑎,𝑏), respectively.

Proof. Without loss of generality, we assume that 𝑎>𝑏. Let 𝑡=𝑎/𝑏>1. Then from (1.1) and (1.2) we have 𝐽₂()()=𝑏𝑡𝑎,𝑏−𝑇𝑎,𝑏²+𝑡+16(𝑡+1)arctan((𝑡−1)/(𝑡+1))4arctan𝑡−1−3𝑡𝑡+1²−1𝑡².+𝑡+1 (3.2) Let 𝑓(𝑡)=4arctan𝑡−1−3𝑡𝑡+1²−1𝑡².+𝑡+1 (3.3) Then simple computations lead to 𝑓(1)𝑓=0,_′(𝑡)=(𝑡−1)⁴𝑡²+1𝑡²+𝑡+1²>0, (3.4) for 𝑡>1.
Therefore, 𝑇(𝑎,𝑏)<𝐽₂(𝑎,𝑏) for all 𝑎,𝑏>0 with 𝑎≠𝑏 follows from (3.2)–(3.4).
Next, we prove that 𝑇(𝑎,𝑏)>𝐽_2/(𝜋−2)(𝑎,𝑏) (3.5) for all 𝑎,𝑏>0 with 𝑎≠𝑏.
Let 𝑝=2/(𝜋−2)=1.75⋯. Then (1.1) and (1.2) lead to 𝑇(𝑎,𝑏)−𝐽_𝑝()=𝑡𝑎,𝑏𝑏𝑝^𝑝+1−12𝑝+1(𝑡^𝑝))/(()𝑡−1arctan((𝑡−1𝑡+1))𝑝+1𝑡−1^𝑝−1𝑝𝑡^𝑝+1−1−2arctan𝑡−1.𝑡+1 (3.6) Let 𝐺(𝑡)=𝑝+1(𝑡−1)𝑡^𝑝−1𝑝𝑡^𝑝+1−1−2arctan𝑡−1.𝑡+1 (3.7) Then simple computations lead to lim_𝑡→1𝐺(𝑡)=lim_𝑡→+∞𝐺(𝑡)𝐺=0,_′(𝑡)=𝑔(𝑡)𝑝𝑡^𝑝+1−1²𝑡²,+1 (3.8)(3.9) where 𝑔(𝑡) is defined as in Lemma 2.1.
From Lemma 2.1 and (3.9) we know that there exists 𝜆>1 such that 𝐺(𝑡) is strictly increasing in [1,𝜆] and strictly decreasing in [𝜆,∞). Then (3.8) leads to that 𝐺(𝑡)>0, (3.10) for 𝑡>1.
Therefore, the inequality (3.5) follows from (3.6), (3.7), and (3.10).
Finally, we prove that 𝐽_2/(𝜋−2)(𝑎,𝑏) and 𝐽₂(𝑎,𝑏) are the best possible lower and upper one-parameter mean bounds for the Seiffert mean 𝑇(𝑎,𝑏), respectively.
Let 𝑝=2/(𝜋−2), 0<𝜀<2 and 𝑥>0. Then from (1.1) and (1.2) one has lim_𝑥→+∞𝐽_𝑝+𝜀(𝑥,1)𝑇()=𝑥,1𝑝+𝜀×𝜋𝑝+𝜀+12>𝑝×𝜋𝑝+12𝑇()=1,1+𝑥,1−𝐽_2−𝜀()=ℎ(𝑥)1+𝑥,12(3−𝜀)(1+𝑥)^2−𝜀((,−1arctan𝑥/2+𝑥)) (3.11)(3.12) where ℎ(𝑥)=(3−𝜀)𝑥(1+𝑥)^2−𝜀−1−2(2−𝜀)(1+𝑥)^3−𝜀𝑥−1arctan.2+𝑥 (3.13)
Letting 𝑥→0 and making use of Taylor expansion we get ℎ(𝑥)=()𝑥)()3−𝜀(2−𝜀𝑥+2−𝜀)(1−𝜀2𝑥²−𝜀(1−𝜀)(2−𝜀)6𝑥³+𝑜𝑥³()()−22−𝜀)(3−𝜀𝑥+3−𝜀)(2−𝜀2𝑥²+(1−𝜀)(2−𝜀)(3−𝜀)6𝑥³+𝑜𝑥³×𝑥2−𝑥²4+𝑥³𝑥12+𝑜³=1𝜀()𝑥122−𝜀)(3−𝜀⁴+𝑜𝑥⁴. (3.14)
The inequality (3.11) implies that for any 0<𝜀<2, there exists 𝑋=𝑋(𝜀)>1, such that 𝑇(𝑥,1)<𝐽_{2/(𝜋−2)+𝜀}(𝑥,1) for 𝑥∈(𝑋,+∞).
Equations (3.12)–(3.14) imply that for any 0<𝜀<2, there exists 𝛿=𝛿(𝜀)>0 such that 𝑇(1+𝑥,1)>𝐽_2−𝜀(1+𝑥,1) for 𝑥∈(0,𝛿).

Acknowledgments

This paper was supported by the Natural Science Foundation of Huzhou Teachers College (Grant no. 2010065) and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant no. Y201016277).