Sharp Bounds for Power Mean in Terms of Generalized Heronian Mean

Abstract

For 1<𝑟<+∞, we find the least value 𝛼 and the greatest value 𝛽 such that the inequality 𝐻_𝛼(𝑎,𝑏)<𝐴_𝑟(𝑎,𝑏)<𝐻_𝛽(𝑎,𝑏) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏. Here, 𝐻_𝜔(𝑎,𝑏) and 𝐴_𝑟(𝑎,𝑏) are the generalized Heronian and the power means of two positive numbers 𝑎 and 𝑏, respectively.

1. Introduction and Statement of Result

For 𝑎,𝑏>0 with 𝑎≠𝑏, the generalized Heronian mean of 𝑎 and 𝑏 is defined by Janous [1] as 𝐻_𝜔(𝑎,𝑏)=⎧⎪⎨⎪⎩𝑎+𝜔√𝑎𝑏+𝑏√𝜔+2,0≤𝜔<+∞,𝑎𝑏,𝜔=+∞.(1.1) If we take 𝜔=1 in (1.1), then we arrive at the classical Heronian mean 𝐻_𝑒(𝑎,𝑏)=𝑎+√𝑎𝑏+𝑏3.(1.2) The domain of definition for the function 𝜔↦𝐻_𝜔(𝑎,𝑏) can be extended to all 𝜔 with 𝜔∈(−2,+∞), that is, 𝐻_𝜔(𝑎,𝑏)=⎧⎪⎨⎪⎩𝑎+𝜔√𝑎𝑏+𝑏√𝜔+2,−2<𝜔<+∞,𝑎𝑏,𝜔=+∞.(1.3) For all fixed 𝑎,𝑏>0, it is easy to derive that 𝜔↦𝐻_𝜔(𝑎,𝑏), −2<𝜔<+∞ is monotonically decreasing, and lim_𝜔→−2+𝐻_𝜔(𝑎,𝑏)=+∞.(1.4)

Let 𝐴_𝑟(𝑎,𝑏)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑎^𝑟+𝑏^𝑟2^1/𝑟√,𝑟≠0,mmai𝑎nx𝑏{,{𝑎𝑎,,𝑏𝑏}},,𝑟𝑟𝑟===+−0,∞∞,,(1.5) denote the power mean of order 𝑟. In particular, the harmonic, geometric, square-root, arithmetic, and root-square means of 𝑎 and 𝑏 are 𝐻(𝑎,𝑏)=𝐴₋₁(𝑎,𝑏)=2𝑎𝐺(𝑎,𝑏)=𝐴₀𝑎+𝑏,𝑁(𝑎,𝑏)=√𝑎𝑏,₁(𝑎,𝑏)=𝐴_1/2(𝑎,𝑏)=√𝑎+√𝑏2²𝐴(𝑎,𝑏)=𝐴₁,(𝑎,𝑏)=𝑎+𝑏𝑆(𝑎,𝑏)=𝐴₂2,(𝑎,𝑏)=𝑎²+𝑏²2.(1.6) It is well known that the power mean of order 𝑟 given in (1.5) is monotonically increasing in 𝑟, then we can write min{𝑎,𝑏}<𝐻(𝑎,𝑏)<𝐺(𝑎,𝑏)<𝑁₁(𝑎,𝑏)<𝐴(𝑎,𝑏)<𝑆(𝑎,𝑏)<max{𝑎,𝑏}.(1.7)

Recently, the inequalities for means have been the subject of intensive research [1–15]. In particular, many remarkable inequalities for the generalized Heronian and power means can be found in the literature [4–9].

In [4], the authors established two sharp inequalities 23𝐺(𝑎,𝑏)+31𝐻(𝑎,𝑏)≥𝐴^−1/31(𝑎,𝑏),3𝐺(𝑎,𝑏)+32𝐻(𝑎,𝑏)≥𝐴^−2/3(𝑎,𝑏).(1.8)

In [5], Long and Chu found the greatest value 𝑝 and the least value 𝑞 such that the double inequality 𝐴_𝑝(𝑎,𝑏)≤𝐴(𝑎,𝑏)^𝛼𝐺(𝑎,𝑏)^𝛽𝐻(𝑎,𝑏)^{1−𝛼−𝛽}≤𝐴_𝑞(𝑎,𝑏)(1.9) holds for all 𝑎,𝑏>0 and 𝛼,𝛽>0 with 𝛼+𝛽<1.

In [6], Shi et al. gave two optimal inequalities 𝐴𝐺^𝛼(𝑎,𝑏)𝐿^1−𝛼(𝑎,𝑏)≤𝐴_(1+2𝛼)/3(𝑎,𝑏),_𝛼(𝑎,𝑏)𝐿^1−𝛼(𝑎,𝑏)≤𝐴_(1−𝛼)/3(𝑎,𝑏),(1.10) for 0<𝛼<1, where 𝐿(𝑎,𝑏)=𝑎−𝑏log𝑎−log𝑏,𝑎≠𝑏,(1.11) is the logarithmic mean for 𝑎,𝑏>0.

In [7], Guan and Zhu obtained sharp bounds for the generalized Heronian mean in terms of the power mean with 𝜔>0. The optimal values 𝛼 and 𝛽 such that 𝐴_𝛼(𝑎,𝑏)≤𝐻_𝜔(𝑎,𝑏)≤𝐴_𝛽(𝑎,𝑏)(1.12) holds in general are (1)in case of 𝜔∈(0,2],𝛼_max=log2/log(𝜔+2) and 𝛽_min=2/(𝜔+2),(2) in case of 𝜔∈[2,+∞),𝛼_max=2/(𝜔+2) and 𝛽_min=log2/log(𝜔+2).

In this paper, we find the least value 𝛼 and the greatest value 𝛽, such that for any fixed 1<𝑟<+∞, the inequality 𝐻_𝛼(𝑎,𝑏)<𝐴_𝑟(𝑎,𝑏)<𝐻_𝛽(𝑎,𝑏)(1.13) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏.

Theorem 1.1. For 1<𝑟<+∞, the optimal numbers 𝛼 and 𝛽 such that 𝐻_𝛼(𝑎,𝑏)<𝐴_𝑟(𝑎,𝑏)<𝐻_𝛽(𝑎,𝑏)(1.14) is valid for all 𝑎,𝑏>0 with 𝑎≠𝑏, are 𝛼_min=2^1/𝑟−2 and 𝛽_max=2(1−𝑟)/𝑟.

Notice that in our case 𝑟>1; the two numbers 𝛼_min and 𝛽_max are all negative see Corollary 2.2 below. Thus, the result in this paper is different from [7, Theorem A].

2. Preliminary Lemmas

The following lemma will be repeatedly used in the proof of Theorem 1.1.

Lemma 2.1. For 1<𝑟<+∞, one has 𝑟2^1/𝑟−1>1.(2.1)

Proof. We show that 𝑚(𝑟)=(1−𝑟)log2+𝑟log𝑟>0,(2.2) which is clearly equivalent to the claim. Equation (2.2) follows from the facts lim_𝑟→1+𝑚(𝑟)=0,𝑚^(𝑟)=−log2+log𝑟+1>0.(2.3)

Corollary 2.2. If 1<𝑟<+∞, then −2<2(1−𝑟)𝑟<2^1/𝑟−2<0.(2.4)

Proof. Since for 1<𝑟<+∞, the two functions 𝜑₁(𝑟)=2(1−𝑟)𝑟,𝜑²(𝑟)=2^1/𝑟−2(2.5) are strictly decreasing, then one has −2=lim_𝑟→+∞𝜑₁(𝑟)<𝜑₁(𝑟),𝜑₂(𝑟)<lim_𝑟→1+𝜑₂(𝑟)=0.(2.6) It suffices to show that 2−2𝑟<𝑟2^1/𝑟−2𝑟,(2.7) which is equivalent to (2.1).

Lemma 2.3. For 𝑥>1 and 𝑟>1, let ℓ(𝑥)=𝑥^2𝑟+1^1/𝑟−2𝑥^2(𝑟−1)𝑥^2𝑟+2𝑟−1.(2.8) Then, ℓ(𝑥) is strictly decreasing for 𝑥>1, and lim_𝑥→1+ℓ(𝑥)=𝑟2^1/𝑟−1,lim_𝑥→+∞ℓ(𝑥)=1.(2.9)

Proof. The fact ℓ(𝑥)>0 for 𝑥>1 and 𝑟>1 is obvious, which allows us to take the logarithmic function of ℓ(𝑥), logℓ(𝑥)=1𝑟−2log𝑥^2𝑟+1+2(𝑟−1)log𝑥+log𝑥^2𝑟+2𝑟−1.(2.10) Some tedious, but not difficult calculations lead to logℓ(𝑥)^=1𝑟−22𝑟𝑥^2𝑟−1=𝑥𝑚^2𝑟(+𝑥)1+2(𝑟𝑥−1)+𝑥^2𝑟2+𝑟𝑥2𝑟−^2𝑟−11𝑥𝑥^2𝑟+1𝑥^2𝑟+2𝑟−1,(2.11) where 𝑚(𝑥)==22((𝑟1−−12)𝑟()2𝑥𝑟^2𝑟−𝑥1)^2𝑟+1−2𝑟𝑥−_2𝑟1+(2𝑟−1)𝑥^2𝑟+1𝑥^2𝑟+2𝑟−1+2𝑟𝑥^2𝑟𝑥^2𝑟+1.(2.12) It is easy to see that 𝑚lim_𝑥→1+𝑚(𝑥)=0,(2.13)_(𝑥)=−4𝑟(𝑟−1)(2𝑟−1)𝑥^2𝑟−1<0.(2.14) Equation (2.14) implies that 𝑚(𝑥) is strictly decreasing for 𝑥>1, which together with (2.13) implies 𝑚(𝑥)<0 for 𝑥>1. Thus, by (2.11), logℓ(𝑥)^<0,(2.15) which implies ℓ^(𝑥)=logℓ(𝑥)^ℓ(𝑥)<0.(2.16) Hence, ℓ(𝑥) is strictly decreasing.
It remains to show (2.9). The first equality in (2.9) is obvious. The second one follows from lim_𝑥→+∞ℓ(𝑥)==lilmim_{𝑥→+∞𝑡→0+}𝑥^2𝑟+1^1/𝑟−2𝑥^2(𝑟−1)𝑥^2𝑟+2𝑟−1(2𝑟−1)𝑡^2𝑟+11+𝑡^2𝑟^{(2𝑟−1)/𝑟}=1.(2.17) This ends the proof of Lemma 2.3.

Lemma 2.4. For 𝑥>1, 𝑟>1, and 𝜔=2^1/𝑟−2, let 𝑓_𝑟(𝑥)=2^1/𝑟𝑥²+𝜔𝑥+1−(𝜔+2)𝑥^2𝑟+1^1/𝑟.(2.18) Then, lim_𝑥→+∞lim_𝑥→+∞𝑓_𝑟(𝑥)=−∞,𝑓^𝑟(𝑥)=2^1/𝑟2^1/𝑟−2.(2.19)

Proof. Simple calculations lead to lim_𝑥→+∞𝑓_𝑟(𝑥)=lim_𝑥→+∞2^1/𝑟𝑥²+𝜔𝑥+1−(𝜔+2)𝑥^2𝑟+1^1/𝑟=lim_𝑡→0+2^1/𝑟𝑡²+𝜔𝑡+1−(𝜔+2)𝑡^2𝑟+1^1/𝑟𝑡²lim_𝑥→+∞=−∞,𝑓^𝑟(𝑥)=lim_𝑥→+∞2^1/𝑟(2𝑥+𝜔)−2(𝜔+2)𝑥^2𝑟+1^1/𝑟=lim_𝑡→0+2^1/𝑟(2+𝜔𝑡)−2(𝜔+2)1+𝑡^2𝑟^{(1−𝑟)/𝑟}=lim_𝑡→0+𝑡2^1/𝑟(2+𝜔𝑡)1+𝑡^2𝑟^{(𝑟−1)/𝑟}−2(𝜔+2)𝑡1+𝑡^2𝑟^{(𝑟−1)/𝑟}=lim_𝑡→0+2^1/𝑟𝜔1+𝑡^2𝑟^{(𝑟−1)/𝑟}+2^(1/𝑟)+1(𝑟−1)(2+𝜔𝑡)1+𝑡^2𝑟^−1/𝑟𝑡^2𝑟−11+𝑡^2𝑟^{(𝑟−1)/𝑟}+2(𝑟−1)1+𝑡^2𝑟^−1/𝑟𝑡^2𝑟=2^1/𝑟𝜔=2^1/𝑟𝜔^1/𝑟−2<0,(2.20) where we have used L'Hospital's law. This ends the proof of Lemma 2.4.

3. Proof of Theorem 1.1

Proof. Firstly, we prove that for 1<𝑟<+∞, 𝐻𝐻_{2(1−𝑟)/𝑟}(𝑎,𝑏)>𝐴_𝑟(𝑎,𝑏),(3.1)_21/𝑟−2(𝑎,𝑏)<𝐴_𝑟(𝑎,𝑏)(3.2) hold true for all 𝑎,𝑏>0 with 𝑎≠𝑏. It is no loss of generality to assume that 𝑎>𝑏>0. Let 𝑥=√𝑏/𝑎>1 and 𝜔∈{2(1−𝑟)/𝑟,2^1/𝑟−2}. In view of Corollary 2.2, −2<𝜔<0. Equations (1.3) and (1.5) lead to 1𝑎𝐻^𝜔(𝑎,𝑏)−𝐴_𝑟(𝑎,𝑏)==𝐻𝑥_𝜔𝑥²,1−𝐴_𝑟𝑥²,1₂+𝜔𝑥+1𝜔+2−𝑥^2𝑟+12^1/𝑟=2^1/𝑟𝑥²+𝜔𝑥+1−(𝜔+2)𝑥^2𝑟+1^1/𝑟=𝑓2^1/𝑟(𝜔+2)_𝑟(𝑥)2^1/𝑟(𝜔+2),(3.3) where 𝑓_𝑟(𝑥) is defined by (2.18). It is easy to see that 𝑓lim_𝑥→1+𝑓_𝑟(𝑥)=0,(3.4)_𝑟(𝑥)=2^1/𝑟(2𝑥+𝜔li)m−_𝑥→1+2(𝜔+2)𝑥^2𝑟+1^1/𝑟−1𝑥^2𝑟−1,(3.5)𝑓^𝑟(𝑥)=0.(3.6) By Lemma 2.3, 𝑓^𝑟(𝑥)==2222^1/𝑟_1/𝑟−(𝜔+2)2(1−𝑟)𝑥^2𝑟+1^1/𝑟−2𝑥^4𝑟−2+(2𝑟−1)𝑥^2𝑟+1^1/𝑟−1𝑥^2(𝑟−1)lim_𝑥→1+−(𝜔+2)ℓ(𝑥)>22^1/𝑟−(𝜔+2)𝑟2^1/𝑟−1=2^1/𝑟[2−(𝜔+2)𝑟],(3.7)𝑓^𝑟(𝑥)=22^1/𝑟−(𝜔+2)2(1−𝑟)2^1/𝑟−2+(2𝑟−1)2^1/𝑟−1=2^1/𝑟[2−(𝜔+2)𝑟].(3.8)
We now distinguish between two cases.
Case 1 (𝜔=2(1−𝑟)/𝑟). Since 2−(𝜔+2)𝑟=0, then by (3.7), 𝑓^𝑟(𝑥)>0. Thus, 𝑓^𝑟(𝑥) is strictly increasing for 𝑥>1, which together with (3.6) implies 𝑓^𝑟(𝑥)>0. Hence, 𝑓_𝑟(𝑥) is strictly increasing for 𝑥>1. Since (3.4), then 𝑓_𝑟(𝑥)>0. Equation (3.1) follows from (3.3).Case 2 (𝜔=2^1/𝑟−2). By (3.5) and (2.11), 𝑓^{𝑟}(𝑥)==−−22((𝜔𝜔++22))ℓ^(𝑥)=𝑚−(𝑥2)(𝜔ℓ(+𝑥)2)logℓ(𝑥)^ℓ(𝑥)>0.𝑥𝑥^2𝑟+1(𝑥^𝑟+2𝑟−1)(3.9) Thus, 𝑓^𝑟(𝑥) is strictly increasing. Equations (3.8) and (2.1) imply lim_𝑥→1+𝑓^𝑟(𝑥)=2^1/𝑟[2−(𝜔+2)𝑟]=2^1/𝑟+11−𝑟2^1/𝑟−1<0.(3.10) Equations (3.7) and (2.9) imply lim_𝑥→+∞𝑓^𝑟(𝑥)=lim_𝑥→+∞22^1/𝑟−(𝜔+2)ℓ(𝑥)=22^1/𝑟−1>0.(3.11) Combining (3.10) with (3.11), we obviously know that there exists 𝜆₁>1 such that 𝑓^𝑟(𝑥)<0 for 𝑥∈(1,𝜆₁) and 𝑓^𝑟(𝑥)>0 for 𝑥∈(𝜆₁,+∞). This implies that 𝑓^𝑟(𝑥) is strictly decreasing for 𝑥∈(1,𝜆₁) and strictly increasing for 𝑥∈(𝜆₁,+∞). By (3.6) and Lemma 2.4, we know that 𝑓^𝑟(𝑥)<0 for 𝑥>1. Therefore, 𝑓_𝑟(𝑥) is strictly decreasing. By (3.4) and Lemma 2.4 again, we derive that 𝑓_𝑟(𝑥)<0 for 𝑥>1. Equation (3.2) follows from (3.3).Secondly, we prove that 𝐻₂^1/𝑟₋₂(𝑎,𝑏) is the best lower bound for the power mean 𝐴_𝑟(𝑎,𝑏) for 1<𝑟<+∞. For any 𝛼<2^1/𝑟−2, lim_𝑥→+∞𝐻_𝛼(𝑥,1)𝐴_𝑟(𝑥,1)=lim^𝑥→+∞2^1/𝑟𝑥+𝛼√𝑥+1(𝛼+2)(𝑥^𝑟+1)^𝑟=2^1/𝑟𝛼+2>1.(3.12) Hence, there exists 𝑋=𝑋(𝛼)>1 such that 𝐻_𝛼(𝑥,1)>𝐴_𝑟(𝑥,1) for 𝑥∈(𝑋,+∞).
Finally, we prove that 𝐻_{2(1−𝑟)/𝑟}(𝑎,𝑏) is the best upper bound for the power mean 𝐴_𝑟(𝑎,𝑏) for 1<𝑟<+∞. For any 𝛽>2(1−𝑟)/𝑟, by (3.7) (with 𝛽 in place of 𝜔), we have lim_𝑥→1+𝑓^𝑟(𝑥)=2^1/𝑟[2−(𝛽+2)𝑟]<0.(3.13) Hence, by the continuity of 𝑓^𝑟(𝑥), there exists 𝛿=𝛿(𝛽)>0 such that 𝑓^𝑟(𝑥)<0 for 𝑥∈(1,1+𝛿). Thus 𝑓_𝑟(𝑥) is strictly decreasing for 𝑥∈(1,1+𝛿). From (3.6), 𝑓^𝑟(𝑥)<0 for 𝑥∈(1,1+𝛿). This result together with (3.4) implies that 𝑓_𝑟(𝑥)<0 for 𝑥∈(1,1+𝛿). Hence, by (3.3), 𝐻_𝛽𝑥²,1<𝐴_𝑟𝑥²,1,(3.14) for 𝑥∈(1,1+𝛿).

Acknowledgments

This work was supported by NSFC (10971224) and NSF of Hebei Province (A2011201011).