Optimal Inequalities for Power Means

Abstract

We present the best possible power mean bounds for the product 𝑀^𝛼_𝑝(𝑎,𝑏)𝑀^1−𝛼_−𝑝(𝑎,𝑏) for any 𝑝>0, 𝛼∈(0,1), and all 𝑎,𝑏>0 with 𝑎≠𝑏. Here, 𝑀_𝑝(𝑎,𝑏) is the 𝑝th power mean of two positive numbers 𝑎 and 𝑏.

1. Introduction

For 𝑝∈ℝ, the 𝑝th power mean 𝑀_𝑝(𝑎,𝑏) of two positive numbers 𝑎 and 𝑏 is defined by 𝑀_𝑝()=𝑎𝑎,𝑏^𝑝+𝑏^𝑝2^1/𝑝√,𝑝≠0,𝑎𝑏,𝑝=0. (1.1)

It is well known that 𝑀_𝑝(𝑎,𝑏) is continuous and strictly increasing with respect to 𝑝∈ℝ for fixed 𝑎,𝑏>0 with 𝑎≠𝑏. Many classical means are special cases of the power mean, for example, 𝑀₋₁(𝑎,𝑏)=𝐻(𝑎,𝑏)=2𝑎𝑏/(𝑎+𝑏), 𝑀₀(𝑎,𝑏)=𝐺(𝑎,𝑏)=√𝑎𝑏 and 𝑀₁(𝑎,𝑏)=𝐴(𝑎,𝑏)=(𝑎+𝑏)/2 are the harmonic, geometric and arithmetic means of 𝑎 and 𝑏, respectively. Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities and properties for the power mean can be found in literature [1–22].

Let 𝐿(𝑎,𝑏)=(𝑎−𝑏)/(log𝑎−log𝑏), 𝑃(𝑎,𝑏)=(𝑎−𝑏)/[4arctan(𝑎/𝑏)−𝜋] and 𝐼(𝑎,𝑏)=1/𝑒(𝑎^𝑎/𝑏^𝑏)^{1/(𝑎−𝑏)} be the logarithmic, Seiffert and identric means of two positive numbers 𝑎 and 𝑏 with 𝑎≠𝑏, respectively. Then it is well known that min{𝑎,𝑏}<𝐻(𝑎,𝑏)<𝐺(𝑎,𝑏)<𝐿(𝑎,𝑏)<𝑃(𝑎,𝑏)<𝐼(𝑎,𝑏)<𝐴(𝑎,𝑏)<max{𝑎,𝑏}, (1.2) for all 𝑎,𝑏>0 with 𝑎≠𝑏.

In [23–29], the authors presented the sharp power mean bounds for 𝐿, 𝐼, (𝐼𝐿)^1/2 and (𝐿+𝐼)/2 as follows: 𝑀₀(𝑎,𝑏)<𝐿(𝑎,𝑏)<𝑀_1/3(𝑎,𝑏),𝑀_2/3(𝑎,𝑏)<𝐼(𝑎,𝑏)<𝑀_log2(),𝑀𝑎,𝑏₀(𝑎,𝑏)<𝐿(𝑎,𝑏)𝐼(𝑎,𝑏)<𝑀_1/2(),1𝑎,𝑏2(𝐿(𝑎,𝑏)+𝐼(𝑎,𝑏))<𝑀_1/2(𝑎,𝑏), (1.3) for all 𝑎,𝑏>0 with 𝑎≠𝑏.

Alzer and Qiu [12] proved that the inequality 12(𝐿(𝑎,𝑏)+𝐼(𝑎,𝑏))>𝑀_𝑝(𝑎,𝑏) (1.4) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝑝≤(log2)/(1+log2)=0.40938….

The following sharp bounds for the sum 𝛼𝐴(𝑎,𝑏)+(1−𝛼)𝐿(𝑎,𝑏), and the products 𝐴^𝛼(𝑎,𝑏)𝐿^1−𝛼(𝑎,𝑏) and 𝐺^𝛼(𝑎,𝑏)𝐿^1−𝛼(𝑎,𝑏) in terms of power means were proved in [5, 8]: 𝑀_{log2/(log2−log𝛼)}(𝑎,𝑏)<𝛼𝐴(𝑎,𝑏)+(1−𝛼)𝐿(𝑎,𝑏)<𝑀_(1+2𝛼)/3(),𝑀𝑎,𝑏₀(𝑎,𝑏)<𝐴^𝛼(𝑎,𝑏)𝐿^1−𝛼(𝑎,𝑏)<𝑀_(1+2𝛼)/3(),𝑀𝑎,𝑏₀(𝑎,𝑏)<𝐺^𝛼(𝑎,𝑏)𝐿^1−𝛼(𝑎,𝑏)<𝑀_(1−𝛼)/3(𝑎,𝑏), (1.5) for any 𝛼∈(0,1) and all 𝑎,𝑏>0 with 𝑎≠𝑏.

In [2, 7] the authors answered the questions: for any 𝛼∈(0,1), what are the greatest values 𝑝₁=𝑝₁(𝛼), 𝑝₂=𝑝₂(𝛼), 𝑝₃=𝑝₃(𝛼), and 𝑝₄=𝑝₄(𝛼), and the least values 𝑞₁=𝑞₁(𝛼), 𝑞₂=𝑞₂(𝛼), 𝑞₃=𝑞₃(𝛼), and 𝑞₄=𝑞₄(𝛼), such that the inequalities 𝑀_𝑝1(𝑎,𝑏)<𝑃^𝛼(𝑎,𝑏)𝐿^1−𝛼(𝑎,𝑏)<𝑀_𝑞1(),𝑀𝑎,𝑏_𝑝2(𝑎,𝑏)<𝐴^𝛼(𝑎,𝑏)𝐺^1−𝛼(𝑎,𝑏)<𝑀_𝑞2(),𝑀𝑎,𝑏_𝑝3(𝑎,𝑏)<𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)<𝑀_𝑞3(),𝑀𝑎,𝑏_𝑝4(𝑎,𝑏)<𝐴^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)<𝑀_𝑞4(𝑎,𝑏), (1.6) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏?

It is the aim of this paper to present the best possible power mean bounds for the product 𝑀^𝛼_𝑝(𝑎,𝑏)𝑀^1−𝛼_−𝑝(𝑎,𝑏) for any 𝑝>0, 𝛼∈(0,1) and all 𝑎,𝑏>0 with 𝑎≠𝑏.

2. Main Result

Theorem 2.1. Let 𝑝>0, 𝛼∈(0,1) and 𝑎,𝑏>0 with 𝑎≠𝑏. Then(1)𝑀_{(2𝛼−1)𝑝}(𝑎,𝑏)=𝑀^𝛼_𝑝(𝑎,𝑏)𝑀^1−𝛼_−𝑝(𝑎,𝑏)=𝑀₀(𝑎,𝑏) for 𝛼=1/2,(2)𝑀_{(2𝛼−1)𝑝}(𝑎,𝑏)>𝑀^𝛼_𝑝(𝑎,𝑏)𝑀^1−𝛼_−𝑝(𝑎,𝑏)>𝑀₀(𝑎,𝑏) for 𝛼>1/2 and 𝑀_{(2𝛼−1)𝑝}(𝑎,𝑏)<𝑀^𝛼_𝑝(𝑎,𝑏)𝑀^1−𝛼_−𝑝(𝑎,𝑏)<𝑀₀(𝑎,𝑏) for 𝛼<1/2, and the bounds 𝑀_{(2𝛼−1)𝑝}(𝑎,𝑏) and 𝑀₀(𝑎,𝑏) for the product 𝑀^𝛼_𝑝(𝑎,𝑏)𝑀^1−𝛼_−𝑝(𝑎,𝑏) in either case are best possible.

Proof. From (1.1) we clearly see that 𝑀_𝑝(𝑎,𝑏) is symmetric and homogenous of degree 1. Without loss of generality, we assume that 𝑏=1, 𝑎=𝑥>1.(1)If 𝛼=1/2, then (1.1) leads to 𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)=1+𝑥^𝑝2^1/𝑝1+𝑥^−𝑝2^−1/𝑝=1+𝑥^𝑝2^1/𝑝2𝑥^𝑝1+𝑥^𝑝1/𝑝=𝑥=𝑀²₀(𝑥,1)=𝑀²_{()𝑝2𝛼−1}(𝑥,1). (2.1)(2)Firstly, we compare the value of 𝑀_{(2𝛼−1)𝑝}(𝑥,1) to the value of 𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1) for 𝛼∈(0,1/2)∪(1/2,1). From (1.1) we have log𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)−log𝑀_{(2𝛼−1)𝑝}()=𝛼𝑥,1𝑝log1+𝑥^𝑝2−1−𝛼𝑝log1+𝑥^−𝑝2−1(2𝛼−1)𝑝log1+𝑥^{(2𝛼−1)𝑝}2. (2.2)
Let𝑓(𝑥)=𝛼𝑝log1+𝑥^𝑝2−1−𝛼𝑝log1+𝑥^−𝑝2−1(2𝛼−1)𝑝log1+𝑥^{(2𝛼−1)𝑝}2, (2.3) then simple computations lead to 𝑓(1)(𝑥)=𝑔(𝑥)=0,𝑓′𝑥(1+𝑥^𝑝)1+𝑥^{(2𝛼−1)𝑝}, (2.4)(2.5) where 𝑔(𝑥)=(𝛼−1)𝑥^2𝛼𝑝+𝛼𝑥^𝑝−𝛼𝑥^{(2𝛼−1)𝑝}𝑔(1)(𝑥)+1−𝛼,=0,𝑔′=𝛼𝑝𝑥^𝑝−1ℎ(𝑥), (2.6)(2.7) where ℎ(𝑥)=2(𝛼−1)𝑥^{(2𝛼−1)𝑝}−(2𝛼−1)𝑥^{2(𝛼−1)𝑝}ℎ(1)(𝑥)()𝑥+1,=0,ℎ′=−2𝑝1−𝛼)(2𝛼−1^{2(𝛼−1)𝑝−1}𝑥^𝑝−1. (2.8)(2.9)
If 𝛼∈(1/2,1), then (2.9) implies that ℎ(𝑥) is strictly decreasing in [1,+∞). Therefore, 𝑀_{(2𝛼−1)𝑝}(𝑥,1)>𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1) follows easily from (2.2)–(2.8) and the monotonicity of ℎ(𝑥).
If 𝛼∈(0,1/2), then (2.9) leads to the conclusion that ℎ(𝑥) is strictly increasing in [1,+∞). Therefore, 𝑀_{(2𝛼−1)𝑝}(𝑥,1)<𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1) follows easily from (2.2)–(2.8) and the monotonicity of ℎ(𝑥).
Secondly, we compare the value of 𝑀₀(𝑥,1) to the value of 𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1). It follows from (1.1) thatlog𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)−log𝑀₀()=𝛼𝑥,1𝑝log1+𝑥^𝑝2−1−𝛼𝑝log1+𝑥^−𝑝2−12log𝑥. (2.10)
Let𝐹(𝑥)=𝛼𝑝log1+𝑥^𝑝2−1−𝛼𝑝log1+𝑥^−𝑝2−12log𝑥, (2.11) then simple computations lead to 𝐹(1)(𝑥)=(𝑥=0,𝐹′2𝛼−1)^𝑝−1𝑥(1+𝑥^𝑝)1+𝑥^{(2𝛼−1)𝑝}. (2.12)(2.13)
If 𝛼∈(1/2,1), then (2.13) implies that 𝐹(𝑥) is strictly increasing in [1,+∞). Therefore, 𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)>𝑀₀(𝑥,1) follows easily from (2.10)–(2.12) and the monotonicity of 𝐹(𝑥).
If 𝛼∈(0,1/2), then (2.13) leads to the conclusion that 𝐹(𝑥) is strictly decreasing in [1,+∞). Therefore, 𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)<𝑀₀(𝑥,1) follows easily from (2.10)–(2.12) and the monotonicity of 𝐹(𝑥).
Next, we prove that the bound 𝑀_{(2𝛼−1)𝑝}(𝑎,𝑏) for the product 𝑀^𝛼_𝑝(𝑎,𝑏)𝑀^1−𝛼_−𝑝(𝑎,𝑏) in either case is best possible.
If 𝛼∈(0,1/2), then for any 𝜖∈(0,(1−2𝛼)𝑝) and 𝑥>0 we have𝑀^𝛼_𝑝(1+𝑥,1)𝑀^1−𝛼_−𝑝(1+𝑥,1)−𝑀_{(2𝛼−1)𝑝+𝜖}()=1+𝑥,11+(1+𝑥)^𝑝2^𝛼/𝑝1+(1+𝑥)^−𝑝2^{(𝛼−1)/𝑝}−1+(1+𝑥)^{(2𝛼−1)𝑝+𝜖}2^{1/[(2𝛼−1)𝑝+𝜖]}. (2.14)
Letting 𝑥→0 and making use of Taylor’s expansion, one has1+(1+𝑥)^𝑝2^𝛼/𝑝1+(1+𝑥)^−𝑝2^{(𝛼−1)/𝑝}−1+(1+𝑥)^{(2𝛼−1)𝑝+𝜖}2^{1/[(2𝛼−1)𝑝+𝜖]}=𝛼1+2𝛼𝑥+𝑝+𝛼−28𝑥²+𝑜𝑥²×1+1−𝛼2()𝑥−1−𝛼𝑝+𝛼+18𝑥²+𝑜𝑥²−11+2()𝑥+2𝛼−1𝑝+𝜖−18𝑥²+𝑜𝑥²=11+2()𝑥+2𝛼−1𝑝−18𝑥²+𝑜𝑥²−11+2()𝑥+2𝛼−1𝑝+𝜖−18𝑥²+𝑜𝑥²𝜖=−8𝑥²+𝑜𝑥². (2.15)
Equations (2.14) and (2.15) imply that for any 𝛼∈(0,1/2) and 𝜖∈(0,(1−2𝛼)𝑝) there exists 𝛿₁=𝛿₁(𝜖)>0, such that 𝑀^𝛼_𝑝(1+𝑥,1)𝑀^1−𝛼_−𝑝(1+𝑥,1)<𝑀_{(2𝛼−1)𝑝+𝜖}(1+𝑥,1) for 𝑥∈(0,𝛿₁).
If 𝛼∈(1/2,1), then for any 𝜖∈(0,(2𝛼−1)𝑝) and 𝑥>0 we have𝑀^𝛼_𝑝(1+𝑥,1)𝑀^1−𝛼_−𝑝(1+𝑥,1)−𝑀_{(2𝛼−1)𝑝−𝜖}()=1+𝑥,11+(1+𝑥)^𝑝2^𝛼/𝑝1+(1+𝑥)^−𝑝2^{(𝛼−1)/𝑝}−1+(1+𝑥)^{(2𝛼−1)𝑝−𝜖}2^{1/[(2𝛼−1)𝑝−𝜖]}. (2.16)
Letting 𝑥→0 and making use of Taylor’s expansion, one has1+(1+𝑥)^𝑝2^𝛼/𝑝1+(1+𝑥)^−𝑝2^{(𝛼−1)/𝑝}−1+(1+𝑥)^{(2𝛼−1)𝑝−𝜖}2^{1/[(2𝛼−1)𝑝−𝜖]}=𝛼1+2𝛼𝑥+𝑝+𝛼−28𝑥²+𝑜𝑥²×1+1−𝛼2()𝑥−1−𝛼𝑝+𝛼+18𝑥²+𝑜𝑥²−11+2()𝑥+2𝛼−1𝑝−𝜖−18𝑥²+𝑜𝑥²=11+2()𝑥+2𝛼−1𝑝−18𝑥²+𝑜𝑥²−11+2()𝑥+2𝛼−1𝑝−𝜖−18𝑥²+𝑜𝑥²=𝜖8𝑥²+𝑜𝑥². (2.17)
Equations (2.16) and (2.17) imply that for any 𝛼∈(1/2,1) and 𝜖∈(0,(2𝛼−1)𝑝) there exists 𝛿₂=𝛿₂(𝜖)>0, such that 𝑀^𝛼_𝑝(1+𝑥,1)𝑀^1−𝛼_−𝑝(1+𝑥,1)>𝑀_{(2𝛼−1)𝑝−𝜖}(1+𝑥,1) for 𝑥∈(0,𝛿₂).
Finally, we prove that the bound 𝑀₀(𝑎,𝑏) for the product 𝑀^𝛼_𝑝(𝑎,𝑏)𝑀^1−𝛼_−𝑝(𝑎,𝑏) in either case is best possible.
If 𝛼∈(0,1/2), then for any 𝜖>0 we clearly see thatlim_𝑥→+∞𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)𝑀_−𝜖(𝑥,1)=+∞. (2.18)
Equation (2.18) implies that for any 𝛼∈(0,1/2) and 𝜖>0 there exists 𝑇₁=𝑇₁(𝜖)>1, such that 𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)>𝑀_−𝜖(𝑥,1) for 𝑥∈(𝑇₁,+∞).
If 𝛼∈(1/2,1), then for any 𝜖>0 we havelim_𝑥→+∞𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)𝑀_𝜖(𝑥,1)=0. (2.19)
Equation (2.19) implies that for any 𝛼∈(1/2,1) and 𝜖>0 there exists 𝑇₂=𝑇₂(𝜖)>1, such that 𝑀^𝛼_𝑝(𝑥,1)𝑀^1−𝛼_−𝑝(𝑥,1)<𝑀_𝜖(𝑥,1) for 𝑥∈(𝑇₂,+∞).

Acknowledgments

This paper was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Hunan Province under Grant 09JJ6003, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.