Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means

Abstract

We present sharp upper and lower generalized logarithmic mean bounds for the geometric weighted mean of the geometric and harmonic means.

1. Introduction

For 𝑝∈ℝ the generalized logarithmic mean 𝐿_𝑝(𝑎,𝑏) of two positive numbers 𝑎 and 𝑏 is defined by 𝐿_𝑝(𝑎,𝑏)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎,𝑎𝑎=𝑏,^𝑝+1−𝑏^𝑝+1(𝑝+1)(𝑎−𝑏)^1/𝑝1,𝑝≠0,𝑝≠−1,𝑎≠𝑏,𝑒𝑎𝑏^𝑏𝑎^{1/(𝑏−𝑎)}𝑏−𝑎,𝑝=0,𝑎≠𝑏,log𝑏−log𝑎,𝑝=−1,𝑎≠𝑏.(1.1)

It is well-known that 𝐿_𝑝(𝑎,𝑏) is continuous and strictly increasing with respect to 𝑝∈ℝ for fixed 𝑎 and 𝑏 with 𝑎≠𝑏. In the recent past, the generalized logarithmic mean has been the subject of intensive research. In particular, many remarkable inequalities for 𝐿_𝑝 can be found in the literature [1–23]. The generalized logarithmic mean has applications in convex function, economics, physics, and even in meteorology [24–27]. In [26] the authors study a variant of Jensen’s functional equation involving 𝐿_𝑝, which appear in a heat conduction problem. Let 𝐴(𝑎,𝑏)=(𝑎+𝑏)/2,𝐼(𝑎,𝑏)=(1/𝑒)(𝑏^𝑏/𝑎^𝑎)^{1/(𝑏−𝑎)}, 𝐿(𝑎,𝑏)=(𝑏−𝑎)/(log𝑏−log𝑎), 𝐺(𝑎,𝑏)=√𝑎𝑏, and 𝐻(𝑎,𝑏)=2𝑎𝑏/(𝑎+𝑏) be the arithmetic, identric, logarithmic, geometric, and harmonic means of two positive numbers 𝑎 and 𝑏 with 𝑎≠𝑏, respectively. Then it is well known thatmin{𝑎,𝑏<}<𝐼(𝐻𝑎,𝑏(𝑎),=𝑏)𝐿<₀𝐺(𝑎,𝑏)=𝐿₋₂(𝑎,𝑏)<𝐿(𝑎,𝑏)=𝐿₋₁(𝑎,𝑏)(𝑎,𝑏)<𝐴(𝑎,𝑏)=𝐿₁(𝑎,𝑏)<max{𝑎,𝑏}.(1.2)

In [28–30], the authors present bounds for 𝐿 and 𝐼 in terms of 𝐺 and 𝐴.

Proposition 1.1. For all positive real numbers 𝑎 and 𝑏 with 𝑎≠𝑏, one has 𝐴^1/3(𝑎,𝑏)𝐺^2/3(𝑎,𝑏)<𝐿(𝑎,𝑏)<113𝐴(𝑎,𝑏)+23𝐺(𝑎,𝑏),3𝐺(𝑎,𝑏)+23𝐴(𝑎,𝑏)<𝐼(𝑎,𝑏).(1.3)

The proof of the following Proposition 1.2 can be found in [31].

Proposition 1.2. For all positive real numbers 𝑎 and 𝑏 with 𝑎≠𝑏, we have √𝐺(𝑎,𝑏)𝐴(𝑎,𝑏)<√𝐿(𝑎,𝑏)𝐼(𝑎,𝑏)<12(𝐿(𝑎,𝑏)+𝐼(𝑎,𝑏))<12(𝐺(𝑎,𝑏)+𝐴(𝑎,𝑏)).(1.4)

For 𝑟∈ℝ the 𝑟th power mean 𝑀_𝑟(𝑎,𝑏) of two positive numbers 𝑎 and 𝑏 is defined by𝑀_𝑟(𝑎,𝑏)=⎧⎪⎨⎪⎩𝑎^𝑟+𝑏^𝑟2^1/𝑟√,𝑟≠0,𝑎𝑏,𝑟=0.(1.5)

The main properties of these means are given in [32]. Several authors discussed the relationship of certain means to 𝑀_𝑟. The following sharp bounds for 𝐿, 𝐼, (𝐼𝐿)^1/2, and (𝐼+𝐿)/2 in terms of power means are proved in [31, 33–37].

Proposition 1.3. For all positive real numbers 𝑎 and 𝑏 with 𝑎≠𝑏 one has 𝑀₀(𝑎,𝑏)<𝐿(𝑀𝑎,𝑏)<𝑀_1/3(𝑎,𝑏),𝑀_2/3(𝑎,𝑏)<𝐼(𝑎,𝑏)<𝑀_log2(𝑎,𝑏),₀(𝑎1,𝑏)<𝐼^1/2(𝑎,𝑏)𝐿^1/2(𝑎,𝑏)<𝑀_1/2(𝑎,𝑏),2[𝐼(𝑎,𝑏)+𝐿(𝑎,𝑏)]<𝑀^1/2(𝑎,𝑏).(1.6)

The following three results were established by Alzer and Qiu in [38].

Proposition 1.4. The inequalities 𝛼𝐴(𝑎,𝑏)+(1−𝛼)𝐺(𝑎,𝑏)<𝐼(𝑎,𝑏)<𝛽𝐴(𝑎,𝑏)+(1−𝛽)𝐺(𝑎,𝑏)(1.7) hold for all positive real numbers 𝑎 and 𝑏 with 𝑎≠𝑏 if and only if 𝛼≤23,𝛽≥2𝑒=0.73575⋅⋅⋅.(1.8)

Proposition 1.5. Let 𝑎 and 𝑏 be real numbers with 𝑎≠𝑏. If 0<𝑎,𝑏≤𝑒, then [𝐺(𝑎,𝑏)]^{𝐴(𝑎,𝑏)}<[𝐿(𝑎,𝑏)]^{𝐼(𝑎,𝑏)}<[𝐴(𝑎,𝑏)]^{𝐺(𝑎,𝑏)}.(1.9) And, if 𝑎,𝑏≥𝑒, then [𝐴(𝑎,𝑏)]^{𝐺(𝑎,𝑏)}<[𝐼(𝑎,𝑏)]^{𝐿(𝑎,𝑏)}<[𝐺(𝑎,𝑏)]^{𝐴(𝑎,𝑏)}.(1.10)

Proposition 1.6. For all positive real numbers 𝑎 and 𝑏 with 𝑎≠𝑏, one has 𝑀_𝑐(𝑎,𝑏)<12(𝐿(𝑎,𝑏)+𝐼(𝑎,𝑏))(1.11) with the best possible parameter 𝑐=log2/(1+log2)=0.40938⋯

In [39] the authors presented inequalities between the generalized logarithmic mean and the product 𝐴^𝛼(𝑎,𝑏)𝐺^𝛽(𝑎,b)𝐻^𝛾(𝑎,𝑏) for all 𝑎,𝑏>0 with 𝑎≠𝑏 and 𝛼,𝛽>0 with 𝛼+𝛽<1.

It is the aim of this paper to give a solution to the problem: for 𝛼∈(0,1), what are the greatest value 𝑝 and the least value 𝑞, such that the inequality𝐿_𝑝(𝑎,𝑏)≤𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)≤𝐿_𝑞(𝑎,𝑏)(1.12) holds for all 𝑎,𝑏>0?

2. Main Result

Theorem 2.1. For 𝛼∈(0,1) and all 𝑎,𝑏>0, one has the following:(1)𝐿_3𝛼−5(𝑎,𝑏)=𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)=𝐿_−(2/𝛼)(𝑎,𝑏) for 𝛼=2/3,(2)𝐿_3𝛼−5(𝑎,𝑏)≥𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)≥𝐿_−(2/𝛼)(𝑎,𝑏) for 0<𝛼<2/3, and 𝐿_3𝛼−5(𝑎,𝑏)≤𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)≤𝐿_−(2/𝛼)(𝑎,𝑏) for 2/3<𝛼<1, with equality if and only if 𝑎=𝑏, and the parameters 3𝛼−5 and −2/𝛼 in each inequality cannot be improved.

Proof. (1) If 𝛼=2/3 and 𝑎=𝑏, then (1.1) implies that 𝐿_3𝛼−5(𝑎,𝑏)=𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)=𝐿_−(2/𝛼)(𝑎,𝑏)=𝑎.
If 𝛼=2/3 and 𝑎≠𝑏, then (1.1) leads to 𝐿_3𝛼−5(𝑎,𝑏)=𝐿_−2/𝛼(𝑎,𝑏)=𝐿₋₃(𝑎,𝑏)=𝑎⁻²−𝑏⁻²2(𝑏−𝑎)^−1/3=(𝑎𝑏)^1/32𝑎𝑏𝑎+𝑏^1/3=𝐺^2/3(𝑎,𝑏)𝐻^1/3(𝑎,𝑏)=𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏).(2.1)
(2) If 𝑎=𝑏, then from (1.1) we clearly see that 𝐿_3𝛼−5(𝑎,𝑏)=𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)=𝐿_−(2/𝛼)(𝑎,𝑏)=𝑎 for any 𝛼∈(0,1).
If 𝑎≠𝑏, without loss of generality, we assume 𝑎>𝑏. Let 𝑎/𝑏=𝑡>1 and 𝑓(𝑡)=log𝐿_3𝛼−5(𝑎,𝑏)−log𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏).(2.2) Then (1.1) and simple computations yield 𝑓(𝑡)=13𝛼−5log𝑡^3𝛼−4−1(3𝛼−4li)m(_𝑡→1+𝑡−1)−𝛼2log𝑡−(1−𝛼)log12+𝑡𝑡,𝑓𝑓(𝑡)=0,(2.3)_(𝑡)=−𝑡^4−3𝛼𝑡𝑡²−1𝑡^4−3𝛼−1𝑔(𝑡),(2.4) where 𝑔3(𝛼𝑡)))𝑡=₂(2−𝛼/2)𝑡^3𝛼−2−((2−𝛼)(2−3𝛼)/5−3𝛼)𝑡^3𝛼−3+((1−𝛼)(2−3𝛼)/2(5−3𝛼))𝑡^3𝛼−4−((1−𝛼)(2−3𝛼)/2(5−+((2−𝛼)(2−3𝛼)/(5−3𝛼))𝑡−(2−𝛼)/2, 𝑔𝑔(1)=0,_(𝑡)=(2−𝛼)(3𝛼−2)2𝑡^3𝛼−3−3(2−𝛼)(2−3𝛼)(𝛼−1)5−3𝛼𝑡^3𝛼−4+(1−𝛼)(2−3𝛼)(3𝛼−4)2(5−3𝛼)𝑡^3𝛼−5−(1−𝛼)(2−3𝛼)+(2−𝛼)(2−3𝛼)(5−3𝛼)𝑡𝑔(5−3𝛼)_,𝑔(1)=0,_(𝑡)=3(2−𝛼)(3𝛼−2)(𝛼−1)2𝑡^3𝛼−4−3(2−𝛼)(2−3𝛼)(𝛼−1)(3𝛼−4)5−3𝛼𝑡^3𝛼−5−(1−𝛼)(2−3𝛼)(3𝛼−4)2𝑡^3𝛼−6−(1−𝛼)(2−3𝛼)𝑔(5−3𝛼),(2.5)_𝑔(1)=0,(2.6)_(𝑡)=32(1−𝛼)(2−𝛼)(4−3𝛼)(3𝛼−2)𝑡^3𝛼−7(𝑡−1)².(2.7)
If 0<𝛼<2/3, then (2.7) implies 𝑔^(𝑡)<0(2.8) for 𝑡>1.
From (2.3)–(2.6) and (2.8) we know that 𝑓(𝑡)>0 for 𝑡>1.
If 2/3<𝛼<1, then (2.7) leads to 𝑔^(𝑡)>0(2.9) for 𝑡>1. Therefore 𝑓(𝑡)<0 for 𝑡>1 follows from (2.3)–(2.6) and (2.9).
Let ℎ(𝑡)=log𝐿_−(2/𝛼)(𝑎,𝑏)−log𝐺^𝛼(𝑎,𝑏)𝐻^1−𝛼(𝑎,𝑏)(2.10) for 𝑡=𝑎/𝑏>1; then (1.1) and elementary calculations lead to ℎ(𝑡)=−𝛼2log𝑡^{(𝛼−2)/𝛼}−1((𝛼−2)/𝛼l)im(𝑡_𝑡→1+−1)−𝛼2log𝑡−(1−𝛼)log12+𝑡𝑡,ℎℎ(𝑡)=0,(2.11)_(𝑡)=−𝑡^{(2−𝛼)/𝛼}𝑡𝑡²−1𝑡^{(2−𝛼)/𝛼}−1𝑣(𝑡),(2.12) where 𝑣(𝑡)=((2−𝛼)/2)𝑡^{(3𝛼−2)/𝛼}+((3𝛼−2)/2)𝑡^{(2𝛼−2)/𝛼}−((3𝛼−2)/2)𝑡−(2−𝛼)/2, 𝑣𝑣(1)=0,_(𝑡)=(2−𝛼)(3𝛼−2)2𝛼𝑡^{(2𝛼−2)/𝛼}+(3𝛼−2)(𝛼−1)𝛼𝑡^{(𝛼−2)/𝛼}−3𝛼−2𝑣2,(2.13)_𝑣(1)=0,(2.14)_(𝑡)=(2−𝛼)(1−𝛼)(2−3𝛼)𝛼²𝑡^−2/𝛼(𝑡−1).(2.15)
If 𝛼∈(0,2/3), then (2.15) implies 𝑣^(𝑡)>0(2.16) for 𝑡>1.
From (2.11)–(2.14) and (2.16) we know that ℎ(𝑡)<0 for 𝑡>1.
If 𝛼∈(2/3,1), then (2.15) leads to 𝑣^(𝑡)<0(2.17) for 𝑡>1. Therefore, ℎ(𝑡)>0 for 𝑡>1 follows from (2.11)–(2.14) and (2.17).
Next, we prove that the parameters −(2/𝛼) and 3𝛼−5 in either case cannot be improved. The proof is divided into two cases.
Case 1 (𝛼∈(0,2/3)). For any 𝜖>0 and 𝑥∈(0,1), from (1.1) one has 𝐺^𝛼(1,1+𝑥)𝐻^1−𝛼(1,1+𝑥)^{5−3𝛼+𝜖}−𝐿_{3𝛼−5−𝜖}(1,1+𝑥)^{5−3𝛼+𝜖}=𝑓¹(𝑥)(1+𝑥/2)^{(1−𝛼)(5−3𝛼+𝜖)}(1+𝑥)^{4−3𝛼+𝜖}−1,(2.18) where 𝑓₁(𝑥)=(1+𝑥)^{(1−𝛼/2)(5−3𝛼+𝜖)}[(1+𝑥)^{4−3𝛼+𝜖}−1]−(4−3𝛼+𝜖)𝑥(1+𝑥)^{4−3𝛼+𝜖}(1+𝑥/2)^{(1−𝛼)(5−3𝛼+𝜖)}.
Let 𝑥→0; making use of the Taylor expansion, we get 𝑓₁(𝑥)=𝜖(4−3𝛼+𝜖)(5−3𝛼+𝜖)24𝑥³+𝑜𝑥³.(2.19)
Equations (2.18) and (2.19) imply that for any 𝛼∈(0,2/3) and 𝜖>0 there exists 𝛿=𝛿(𝜖,𝛼)∈(0,1), such that 𝐿_{3𝛼−5−𝜖}(1,1+𝑥)<𝐺^𝛼(1,1+𝑥)𝐻^1−𝛼(1,1+𝑥) for 𝑥∈(0,𝛿).
On the other hand, for any 𝜖∈(0,(2/𝛼)−1) we have 𝐿=𝑡_{−(2/𝛼)+𝜖}(1,𝑡)_{𝛼/(2−𝜖𝛼)}−𝐺^𝛼(1,𝑡)𝐻^1−𝛼(1,𝑡)(2/𝛼1−−𝜖−𝑡1)(1−^{−2/𝛼+𝜖+1}1/𝑡)^{−𝛼/(2−𝜖𝛼)}−𝑡^{−𝜖𝛼2/2(2−𝜖𝛼)}2𝑡lim_𝑡→+∞1+𝑡^1−𝛼,(2/𝛼1−−𝜖−𝑡1)(1−^{−2/𝛼+𝜖+1}1/𝑡)^{−𝛼/(2−𝜖𝛼)}−𝑡^{−𝜖𝛼2/2(2−𝜖𝛼)}2𝑡=21+𝑡^1−𝛼𝛼−𝜖−1^{𝛼/(2−𝜖𝛼)}>0.(2.20)
From (2.20) we know that for any 𝛼∈(0,2/3) and 𝜖∈(0,2/𝛼−1) there exists 𝑇=𝑇(𝜖,𝛼)>1, such that 𝐿_{−2/𝛼+𝜖}(1,𝑡)>𝐺^𝛼(1,𝑡)𝐻^1−𝛼(1,𝑡) for 𝑡∈(𝑇,∞).Case 2 (𝛼∈(2/3,1)). For any 𝜖∈(0,4−3𝛼) and 𝑥∈(0,1), from (1.1) one has 𝐿_{3𝛼−5+𝜖}(1,1+𝑥)^{5−3𝛼−𝜖}−𝐺^𝛼(1,1+𝑥)𝐻^1−𝛼(1,1+𝑥)^{5−3𝛼−𝜖}=𝑓²(𝑥)(1+𝑥/2)^{(1−𝛼)(5−3𝛼−𝜖)}(1+𝑥)^{4−3𝛼−𝜖}−1,(2.21) where 𝑓₂(𝑥)=(4−3𝛼−𝜖)𝑥(1+𝑥)^{4−3𝛼−𝜖}(1+𝑥/2)^{(1−𝛼)(5−3𝛼−𝜖)}−(1+𝑥)^{(1−𝛼/2)(5−3𝛼−𝜖)}[(1+𝑥)^{4−3𝛼−𝜖}−1].
Let 𝑥→0; making use of the Taylor expansion, we have 𝑓₂(𝑥)=𝜖24(4−3𝛼−𝜖)(5−3𝛼−𝜖)𝑥³+𝑜𝑥³.(2.22)
Equations (2.21) and (2.22) imply that for any 𝛼∈(2/3,1) and 𝜖∈(0,4−3𝛼) there exists 𝛿=𝛿(𝜖,𝛼)∈(0,1), such that 𝐿_{3𝛼−5+𝜖}(1,1+𝑥)>𝐺^𝛼(1,1+𝑥)𝐻^1−𝛼(1,1+𝑥) for 𝑥∈(0,𝛿).
On the other hand, for any 𝜖>0, we have 𝐺^𝛼(1=,𝑡𝑡)𝐻^1−𝛼_𝛼/2(1,𝑡)−𝐿_{−(2/𝛼)−𝜖}(1,𝑡)12+𝑡𝑡^1−𝛼−𝑡^{−𝜖𝛼2/2(2+𝜖𝛼)}lim_𝑡→+∞(2/𝛼1+−𝜖𝑡−1)(1−^{−(2/𝛼+𝜖−1)}1/𝑡)^{−𝛼/(2+𝜖𝛼)},12+𝑡𝑡^1−𝛼−𝑡^{−𝜖𝛼2/2(2+𝜖𝛼)}(2/𝛼1+−𝜖𝑡−1)(1−^{−(2/𝛼+𝜖−1)}1/𝑡)^{−𝛼/(2+𝜖𝛼)}=2^1−𝛼>0.(2.23)
From (2.23) we know that for any 𝛼∈(2/3,1) and 𝜖>0 there exists 𝑇=𝑇(𝜖,𝛼)>1, such that 𝐿_{−(2/𝛼)−𝜖}(1,𝑡)<𝐺^𝛼(1,𝑡)𝐻^1−𝛼(1,𝑡) for 𝑡∈(𝑇,∞).

Acknowledgment

This work was supported by the Natural Science Foundation of Zhejiang Broad-cast and TV University under Grant XKT-09G21.