A Nice Separation of Some Seiffert-Type Means by Power Means

Abstract

Seiffert has defined two well-known trigonometric means denoted by 𝒫 and 𝒯. In a similar way it was defined by Carlson the logarithmic mean ℒ as a hyperbolic mean. Neuman and Sándor completed the list of such means by another hyperbolic mean ℳ. There are more known inequalities between the means 𝒫,𝒯, and ℒ and some power means 𝒜_𝑝. We add to these inequalities two new results obtaining the following nice chain of inequalities 𝒜₀<ℒ<𝒜_1/2<𝒫<𝒜₁<ℳ<𝒜_3/2<𝒯<𝒜₂, where the power means are evenly spaced with respect to their order.

1. Means

A mean is a function 𝑀∶ℝ²₊→ℝ₊, with the property min(𝑎,𝑏)≤𝑀(𝑎,𝑏)≤max(𝑎,𝑏),∀𝑎,𝑏>0. (1.1) Each mean is reflexive; that is, 𝑀(𝑎,𝑎)=𝑎,∀𝑎>0. (1.2) This is also used as the definition of 𝑀(𝑎,𝑎).

We will refer here to the following means:(i)the power means 𝒜_𝑝, defined by 𝒜_𝑝()=𝑎𝑎,𝑏^𝑝+𝑏^𝑝2^1/𝑝,𝑝≠0; (1.3)(ii)the geometric mean 𝒢, defined as 𝒢(𝑎,𝑏)=√𝑎𝑏, but verifying also the property lim_𝑝→0𝒜_𝑝(𝑎,𝑏)=𝒜₀(𝑎,𝑏)=𝒢(𝑎,𝑏); (1.4)(iii)the first Seiffert mean 𝒫, defined in [1] by 𝒫(𝑎,𝑏)=𝑎−𝑏2sin⁻¹((𝑎−𝑏)/(𝑎+𝑏)),𝑎≠𝑏; (1.5)(iv)the second Seiffert mean 𝒯, defined in [2] by 𝒯(𝑎,𝑏)=𝑎−𝑏2tan⁻¹((𝑎−𝑏)/(𝑎+𝑏)),𝑎≠𝑏; (1.6)(v)the Neuman-Sándor mean ℳ, defined in [3] by ℳ(𝑎,𝑏)=𝑎−𝑏2sinh⁻¹((𝑎−𝑏)/(𝑎+𝑏)),𝑎≠𝑏; (1.7)(vi)the Stolarsky means 𝒮_𝑝,𝑞 defined in [4] as follows: 𝒮_𝑝,𝑞()=𝑞𝑎𝑎,𝑏^𝑝−𝑏^𝑝𝑝(𝑎^𝑞−𝑏^𝑞)^{1/(𝑝−𝑞)}1,𝑝𝑞𝑝−𝑞≠0𝑒^𝑝𝑎^𝑎𝑝𝑏^{𝑏𝑝1/(𝑎𝑝−𝑏𝑝)}𝑎,𝑝=𝑞≠0_𝑝−𝑏^𝑝𝑝(ln𝑎−ln𝑏)^1/𝑝√,𝑝≠0,𝑞=0𝑎𝑏,𝑝=𝑞=0. (1.8)

The mean 𝒜₁=𝒜 is the arithmetic mean and the mean 𝒮_1,0=ℒ is the logarithmic mean. As Carlson remarked in [5], the logarithmic mean can be represented also by ℒ(𝑎,𝑏)=𝑎−𝑏2tanh⁻¹)/(;((𝑎−𝑏𝑎+𝑏)) (1.9) thus the means 𝒫,𝒯,ℳ, and ℒ are very similar. In [3] it is also proven that these means can be defined using the nonsymmetric Schwab-Borchardt mean 𝒮ℬ given by ()=√𝒮ℬ𝑎,𝑏𝑏²−𝑎²cos⁻¹()√𝑎/𝑏,if𝑎<𝑏𝑎²−𝑏²cosh⁻¹(𝑎/𝑏),if𝑎>𝑏 (1.10) (see [6, 7]). It has been established in [3] that ℒ=𝒮ℬ(𝒜,𝒢),𝒫=𝒮ℬ(𝒢,𝒜),𝒯=𝒮ℬ𝒜,𝒜₂,ℳ=𝒮ℬ𝒜₂,𝒜. (1.11)

2. Interlacing Property of Power Means

Given two means 𝑀 and 𝑁, we will write 𝑀<𝑁 if 𝑀(𝑎,𝑏)<𝑁(𝑎,𝑏),for𝑎≠𝑏. (2.1)

It is known that the family of power means is an increasing family of means, thus 𝒜_𝑝<𝒜_𝑞,if𝑝<𝑞. (2.2) Of course, it is more difficult to compare two Stolarsky means, each depending on two parameters. To present the comparison theorem given in [8, 9], we have to give the definitions of the following two auxiliary functions: 𝑘=|𝑥|−𝑦𝑥,𝑦(𝑥)𝑙=ℒ𝑥−𝑦,𝑥≠𝑦sign,𝑥=𝑦,𝑥,𝑦𝑥,𝑦,𝑥>0,𝑦>00,𝑥≥0,𝑦≥0,𝑥𝑦=0. (2.3)

Theorem 2.1. Let 𝑝,𝑞,𝑟,𝑠∈ℝ. Then the comparison inequality 𝒮_𝑝,𝑞≤𝒮_𝑟,𝑠 (2.4) holds true if and only if 𝑝+𝑞≤𝑟+𝑠, and (1) 𝑙(𝑝,𝑞)≤𝑙(𝑟,𝑠) if 0≤min(𝑝,𝑞,𝑟,𝑠), (2) 𝑘(𝑝,𝑞)≤𝑘(𝑟,𝑠) if min(𝑝,𝑞,𝑟,𝑠)<0<max(𝑝,𝑞,𝑟,𝑠), or (3) −𝑙(−𝑝,−𝑞)≤−𝑙(−𝑟,−𝑠) if max(𝑝,𝑞,𝑟,𝑠)≤0.

We need also in what follows an important double-sided inequality proved in [3] for the Schwab-Borchardt mean: ³√𝑎𝑏²<𝒮ℬ(𝑎,𝑏)<𝑎+2𝑏3,𝑎≠𝑏. (2.5)

Being rather complicated, the Seiffert-type means were evaluated by simpler means, first of all by power means. The evaluation of a given mean 𝑀 by power means assumes the determination of some real indices 𝑝 and 𝑞 such that 𝒜_𝑝<𝑀<𝒜_𝑞. The evaluation is optimal if 𝑝 is the the greatest and 𝑞 is the smallest index with this property. This means that 𝑀 cannot be compared with 𝒜_𝑟 if 𝑝<𝑟<𝑞.

For the logarithmic mean in [10], it was determined the optimal evaluation 𝒜₀<𝐿<𝒜_1/3. (2.6) For the Seiffert means, there are known the evaluations 𝒜_1/3<𝑃<𝒜_2/3, (2.7) proved in [11] and 𝒜₁<𝑇<𝒜₂, (2.8) given in [2]. It is also known that 𝒜₁<𝑀<𝒯, (2.9) as it was shown in [3]. Moreover in [12] it was determined the optimal evaluation 𝒜_ln2/ln𝜋<𝑃<𝒜_2/3. (2.10) Using these results we deduce the following chain of inequalities: 𝒜₀<𝐿<𝒜_1/2<𝑃<𝒜₁<𝑀<𝒯<𝒜₂. (2.11) To prove the full interlacing property of power means, our aim is to show that 𝒜_3/2 can be put between ℳ and 𝒯. We thus obtain a nice separation of these Seiffert-type means by power means which are evenly spaced with respect to their order.

3. Main Results

We add to the inequalities (2.11) the next results.

Theorem 3.1. The following inequalities ℳ<𝒜_3/2<𝑇 (3.1) are satisfied.

Proof. First of all, let us remark that 𝒜_3/2=𝒮_3,3/2. So, for the first inequality in (3.1), it is sufficient to prove that the following chain of inequalities 𝒜ℳ<²+2𝒜3<𝒮_3,1<𝒮_3,3/2 (3.2) is valid. The first inequality in (3.2) is a simple consequence of the property of the mean ℳ given in (1.11) and the second inequality from (2.5). The second inequality can be proved by direct computation or by taking 𝑎=1+𝑡,𝑏=1−𝑡,(0<𝑡<1) which gives √1+𝑡²+23<3+𝑡²3, (3.3) which is easy to prove. The last inequality in (3.2) is given by the comparison theorem of the Stolarsky means. In a similar way, the second inequality in (3.1) is given by the relations 𝒮_3,3/2<𝒮_4,1=³𝒜𝒜²₂<𝑇. (3.4) The first inequality is again given by the comparison theorem of the Stolarsky means. The equality in (3.4) is shown by elementary computations, and the last inequality is a simple consequence of the property of the mean 𝒯 given in (1.11) and the first inequality from (2.5).

Corollary 3.2. The following two-sided inequality 𝑥sinh⁻¹𝑥<𝒜_3/2()<𝑥1−𝑥,1+𝑥tan⁻¹𝑥, (3.5) is valid for all 0<𝑥<1.

Acknowledgment

The authors wish to thank the anonymous referee for offering them a simpler proof for their results.