Optimal Bounds for Neuman-Sándor Mean in Terms of the Convex Combinations of Harmonic, Geometric, Quadratic, and Contraharmonic Means

Abstract

We present the best possible lower and upper bounds for the Neuman-Sándor mean in terms of the convex combinations of either the harmonic and quadratic means or the geometric and quadratic means or the harmonic and contraharmonic means.

1. Introduction

For 𝑎,𝑏>0 with 𝑎≠𝑏, the Neuman-Sándor mean 𝑀(𝑎,𝑏) [1] is defined by 𝑀(𝑎,𝑏)=𝑎−𝑏2sinh⁻¹)/(,[(𝑎−𝑏𝑎+𝑏)] (1.1) where sinh⁻¹(𝑥)=log(𝑥+√1+𝑥²) is the inverse hyperbolic sine function.

Recently, the theory of bivariate means have been the subject of intensive research [2–17]. In particular, many remarkable inequalities for the Neuman-Sándor mean 𝑀(𝑎,𝑏) can be found in the literature [1, 18–20].

Let 𝐻(𝑎,𝑏)=2𝑎𝑏/(𝑎+𝑏), 𝐺(𝑎,𝑏)=√𝑎𝑏, 𝐿(𝑎,𝑏)=(𝑏−𝑎)/(log𝑏−log𝑎), 𝑃(𝑎,𝑏)=(𝑎−𝑏)/(4arctan𝑎/𝑏−𝜋), 𝐴(𝑎,𝑏)=(𝑎+𝑏)/2, 𝑇(𝑎,𝑏)=(𝑎−𝑏)/[2arctan((𝑎−𝑏)/(𝑎+𝑏))], 𝑄(𝑎,𝑏)=(𝑎²+𝑏²)/2 and 𝐶(𝑎,𝑏)=(𝑎²+𝑏²)/(𝑎+𝑏) be the harmonic, geometric, logarithmic, first Seiffert, arithmetic, second Seiffert, quadratic, and contraharmonic means of 𝑎 and 𝑏, respectively. Then it is well known that the inequalities 𝐻()()()()()()()()()𝑎,𝑏<𝐺𝑎,𝑏<𝐿𝑎,𝑏<𝑃𝑎,𝑏<𝐴𝑎,𝑏<𝑀𝑎,𝑏<𝑇𝑎,𝑏<𝑄𝑎,𝑏<𝐶𝑎,𝑏 (1.2) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏.

In [1, 18], Neuman and Sándor proved that the double inequalities 𝐴()()(),𝑃()𝑀()𝑎,𝑏<𝑀𝑎,𝑏<𝑇𝑎,𝑏𝑎,𝑏𝑎,𝑏<𝐴²(),𝐴()𝑇()𝑎,𝑏𝑎,𝑏𝑎,𝑏<𝑀²()<𝐴𝑎,𝑏²(𝑎,𝑏)+𝑇²(𝑎,𝑏)2 (1.3) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏.

Let 0<𝑎, 𝑏<1/2 with 𝑎≠𝑏, 𝑎^′=1−𝑎 and 𝑏^′=1−𝑏. Then the following Ky Fan inequalities 𝐺(𝑎,𝑏)𝐺𝑎^′,𝑏^′<𝐿(𝑎,𝑏)𝐿𝑎^′,𝑏^′<𝑃(𝑎,𝑏)𝑃𝑎^′,𝑏^′<𝐴(𝑎,𝑏)𝐴𝑎^′,𝑏^′<𝑀(𝑎,𝑏)𝑀𝑎^′,𝑏^′<𝑇(𝑎,𝑏)𝑇𝑎^′,𝑏^′ (1.4) were presented in [1].

The double inequality 𝐿_𝑝0(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝐿₂(𝑎,𝑏) for all 𝑎,𝑏>0 with 𝑎≠𝑏 was established by Li et al. in [19], where 𝐿_𝑝(𝑎,𝑏)=[(𝑏^𝑝+1−𝑎^𝑝+1)/((𝑝+1)(𝑏−𝑎))]^1/𝑝 (𝑝≠−1,0), 𝐿₀(𝑎,𝑏)=1/𝑒(𝑏^𝑏/𝑎^𝑎)^{1/(𝑏−𝑎)} and 𝐿₋₁(𝑎,𝑏)=(𝑏−𝑎)/(log𝑏−log𝑎) is the 𝑝th generalized logarithmic mean of 𝑎 and 𝑏, and 𝑝₀=1.843… is the unique solution of the equation (𝑝+1)^1/𝑝=2log(1+√2).

Neuman [20] proved that the double inequalities ()+()𝐴()()()+𝐴(),()+()𝐴()()()+𝐴()𝛼𝑄𝑎,𝑏1−𝛼𝑎,𝑏<𝑀𝑎,𝑏<𝛽𝑄𝑎,𝑏1−𝛽𝑎,𝑏𝜆𝑄𝑎,𝑏1−𝜆𝑎,𝑏<𝑀𝑎,𝑏<𝜇𝑄𝑎,𝑏1−𝜇𝑎,𝑏 (1.5) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼≤1−log(√2+1)/[(√2−1)log(√2+1)]=0.3249…, 𝛽≥1/3, 𝜆≤1−log(√2+1)/log(√2+1)=0.1345… and 𝜇≥1/6.

The main purpose of this paper is to find the least values 𝛼₁, 𝛼₂, 𝛼₃, and the greatest values 𝛽₁, 𝛽₂, 𝛽₃, such that the double inequalities 𝛼₁𝐻(𝑎,𝑏)+1−𝛼₁𝑄(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝛽₁𝐻(𝑎,𝑏)+1−𝛽₁𝑄(),𝛼𝑎,𝑏₂𝐺(𝑎,𝑏)+1−𝛼₂𝑄(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝛽₂𝐺(𝑎,𝑏)+1−𝛽₂𝑄(),𝛼𝑎,𝑏₃𝐻(𝑎,𝑏)+1−𝛼₃𝐶(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝛽₃𝐻(𝑎,𝑏)+1−𝛽₃𝐶(𝑎,𝑏) (1.6) hold true for all 𝑎,𝑏>0 with 𝑎≠𝑏.

Our main results are presented in Theorems 1.1–1.3.

Theorem 1.1. The double inequality 𝛼₁𝐻(𝑎,𝑏)+1−𝛼₁𝑄(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝛽₁𝐻(𝑎,𝑏)+1−𝛽₁𝑄(𝑎,𝑏) (1.7) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼₁≥2/9=0.2222… and 𝛽₁≤1−1/[√2log(1+√2)]=0.1977….

Theorem 1.2. The double inequality 𝛼₂𝐺(𝑎,𝑏)+1−𝛼₂𝑄(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝛽₂𝐺(𝑎,𝑏)+1−𝛽₂𝑄(𝑎,𝑏) (1.8) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼₂≥1/3=0.3333… and 𝛽₂≤1−1/[√2log(1+√2)]=0.1977….

Theorem 1.3. The double inequality 𝛼₃𝐻(𝑎,𝑏)+1−𝛼₃𝐶(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝛽₃𝐻(𝑎,𝑏)+1−𝛽₃𝐶(𝑎,𝑏) (1.9) holds for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼₃≥1−1/[2log(1+√2)]=0.4327… and 𝛽₃≤5/12=0.4166….

2. Lemmas

In order to prove our main results we need two Lemmas, which we present in this section.

Lemma 2.1 (see [21, Lemma 1.1]). Suppose that the power series 𝑓(𝑥)=∑^∞_𝑛=0𝑎_𝑛𝑥^𝑛 and 𝑔(𝑥)=∑^∞_𝑛=0𝑏_𝑛𝑥^𝑛 have the radius of convergence 𝑟>0 and 𝑏_𝑛>0 for all 𝑛∈{0,1,2,…}. Let ℎ(𝑥)=𝑓(𝑥)/𝑔(𝑥), then the following statements are true.(1)If the sequence {𝑎_𝑛/𝑏_𝑛}^∞_𝑛=0 is (strictly) increasing (decreasing), then ℎ(𝑥) is also (strictly) increasing (decreasing) on (0,𝑟).(2)If the sequence {𝑎_𝑛/𝑏_𝑛} is (strictly) increasing (decreasing) for 0<𝑛≤𝑛₀ and (strictly) decreasing (increasing) for 𝑛>𝑛₀, then there exists 𝑥₀∈(0,𝑟) such that ℎ(𝑥) is (strictly) increasing (decreasing) on (0,𝑥₀) and (strictly) decreasing (increasing) on (𝑥₀,𝑟).

Lemma 2.2. Let 𝑝∈(0,1), 𝜆₀=1−1/[√2log(1+√2)]=0.1977… and 𝑓_𝑝(𝑥)=sinh⁻¹(𝑥)−𝑥√1+𝑥²−𝑝√1+𝑥²−√1−𝑥². (2.1) Then 𝑓_1/3(𝑥)<0 and 𝑓_𝜆0(𝑥)>0 for all 𝑥∈(0,1).

Proof. From (2.1), one has 𝑓_𝑝(0)𝑓=0,_𝑝(1)=log1+√2−1√2,𝑓1−𝑝_′𝑝(𝑥)=𝑔_𝑝(𝑥)√1−𝑥⁴√1+𝑥²+𝑝√1−𝑥²−√1+𝑥²², (2.2)(2.3)(2.4) where 𝑔_𝑝(𝑥)=√1−𝑥²√1+𝑥²+𝑝√1−𝑥²−√1+𝑥²²−√1−𝑥²−𝑝√1+𝑥²−√1−𝑥². (2.5)
We divide the proof into two cases.
Case 1. (𝑝=1/3). Then (2.5) leads to 𝑔_1/3(0)=0,𝑔_1/3(1)√=−23𝑔<0,_′1/3(𝑥)=𝑥³√1−𝑥⁴ℎ_1/3(𝑥), (2.6)(2.7) where ℎ_1/3(𝑥)=149√1+𝑥²+√1−𝑥²−√1+𝑥²+√1−𝑥²−√1−𝑥²3. (2.8) We clearly see that the function √1+𝑥²+√1−𝑥² is strictly decreasing in (0,1). Then from (2.8), we get ℎ_1/3(𝑥)<ℎ_1/3(1)2√=−29<0 (2.9) for 𝑥∈(0,1).
Therefore, 𝑓_1/3(𝑥)<0 for all 𝑥∈(0,1) follows easily from (2.2), (2.4), (2.6), (2.7), and (2.9).
Case 2. (𝑝=𝜆₀). Then (2.3) and (2.5) yield 𝑓_𝜆0(1)=𝑔_𝜆0(0)=0,𝑔_𝜆0(1)=−√2𝜆₀𝑔<0,_′𝜆0(𝑥)=𝑥√1−𝑥⁴ℎ_𝜆0(𝑥), (2.10)(2.11) where ℎ_𝜆0(𝑥)=2−3𝜆₀−2𝜆²₀−3−6𝜆₀𝑥²√1+𝑥²−3𝜆₀−2𝜆²₀+6𝜆₀−6𝜆²₀𝑥²√1−𝑥². (2.12) We divide the discussion of this case into two subcases, and all computations are carried out using MATHEMATICA software.
Subcase A. 𝑥∈(0.9,1). Then from (2.12) and the fact that 2−3𝜆₀−2𝜆²₀−3−6𝜆₀𝑥²<2−3𝜆₀−2𝜆²₀−3−6𝜆₀×(0.9)²=−0.1404⋯<0, (2.13) we know that ℎ_𝜆0(𝑥)<0 (2.14)
for 𝑥∈(0.9,1).
Subcase B. 𝑥∈(0,0.9]. Then from (2.12), one has ℎ_𝜆0(0)=0.8137⋯>0,ℎ_𝜆0()ℎ0.9=−0.7494⋯<0,_′𝜆0(𝑥)=𝑥√1−𝑥⁴𝜇(𝑥), (2.15) where 𝜇(𝑥)=18𝜆₀−18𝜆²₀𝑥²−9𝜆₀−10𝜆²₀√1+𝑥²−9−18𝜆₀𝑥²+4−9𝜆₀+2𝜆²₀√1−𝑥². (2.16) We conclude that 𝜇(𝑥)<0 (2.17) for all 𝑥∈(0,0.9]. Indeed, if 𝑥∈(0,1/2), then (2.17) follows from (2.16) and the inequality 18𝜆₀−18𝜆²₀𝑥²−9𝜆₀−10𝜆²₀<5.5𝜆²₀−4.5𝜆₀=−0.6747⋯<0. (2.18) If 𝑥∈[1/2,0.9], then (2.17) follows from (2.16) and the inequalities 18𝜆₀−18𝜆²₀𝑥²−9𝜆₀−10𝜆²₀≤18𝜆₀−18𝜆²₀×(0.9)²−9𝜆₀−10𝜆²₀=5.58𝜆₀−4.58𝜆²₀=0.9242…,9−18𝜆₀𝑥²+4−9𝜆₀+2𝜆²₀≥149−18𝜆₀+4−9𝜆₀+2𝜆²₀=6.25−13.5𝜆₀+2𝜆²₀=3.6589…,18𝜆₀−18𝜆²₀𝑥²−9𝜆₀−10𝜆²₀√1+𝑥²−9−18𝜆₀𝑥²+4−9𝜆₀+2𝜆²₀√1−𝑥²≤5.58𝜆₀−4.58𝜆²₀1+(0.9)²−6.25−13.5𝜆₀+2𝜆²₀1−(0.9)²=−0.3514⋯<0. (2.19) From (2.15) together with (2.17) we clearly see that there exists 𝑥₀∈(0,0.9) such that ℎ_𝜆0(𝑥)>0 for 𝑥∈[0,𝑥₀) and ℎ_𝜆0(𝑥)<0 for (𝑥₀,0.9].
Subcases A and B lead to the conclusion that ℎ_𝜆0(𝑥)>0 for 𝑥∈[0,𝑥₀) and ℎ_𝜆0(𝑥)<0 for 𝑥∈(𝑥₀,1). Thus from (2.11), we know that 𝑔_𝜆0(𝑥) is strictly increasing in (0,𝑥₀] and strictly decreasing in [𝑥₀,1).
It follows from (2.4) and (2.10) together with the piecewise monotonicity of 𝑔_𝜆0(𝑥) that there exists 𝑥₁∈(0,1) such that 𝑓_𝜆0(𝑥) is strictly increasing in [0,𝑥₁) and strictly decreasing in [𝑥₁,1).
Therefore, 𝑓_𝜆0(𝑥)>0 for 𝑥∈(0,1) follows from (2.2) and (2.10) together with the piecewise monotonicity of 𝑓_𝜆0(𝑥).

3. Proof of Theorems 1.1–1.3

Proof of Theorem 1.1. Since 𝐻(𝑎,𝑏), 𝑀(𝑎,𝑏) and 𝑄(𝑎,𝑏) are symmetric and homogeneous of degree 1. Hence, without loss of generality, we assume that 𝑎>𝑏. Let 𝑥=(𝑎−𝑏)/(𝑎+𝑏) and 𝑡=sinh⁻¹(𝑥). Then 𝑥∈(0,1), 𝑡∈(0,log(1+√2)), 𝑀(𝑎,𝑏)/𝐴(𝑎,𝑏)=𝑥/sinh⁻¹(𝑥)=sinh(𝑡)/𝑡, 𝐻(𝑎,𝑏)/𝐴(𝑎,𝑏)=1−𝑥²=1−sinh²(𝑡)=[3−cosh(2𝑡)]/2, 𝑄(𝑎,b)/𝐴(𝑎,𝑏)=√1+𝑥²=cosh(𝑡) and 𝑄(𝑎,𝑏)−𝑀(𝑎,𝑏)𝑄()()=√𝑎,𝑏−𝐻𝑎,𝑏1+𝑥²sinh⁻¹(𝑥)−𝑥√1+𝑥²−1−𝑥²sinh⁻¹(𝑥)=𝑡cosh(𝑡)−sinh(𝑡)𝑡)()(𝑡)−((𝑡).[(1/2cosh2𝑡+cosh3/2)]∶=𝜑 (3.1)
Making use of power series sinh(𝑡)=∑^∞_𝑛=0𝑡^2𝑛+1/(2𝑛+1)! and cosh(𝑡)=∑^∞_𝑛=0𝑡^2𝑛/(2𝑛)!, we can express (3.1) as follows: 𝜑(𝑡)=∑^∞_𝑛=1[2𝑛/((2𝑛+1)(2𝑛)!)]𝑡^2𝑛+1∑^∞_𝑛=12^2𝑛−1+1/(2𝑛)!𝑡^2𝑛+1. (3.2) Let 𝑎_𝑛=2𝑛/((2𝑛+1)(2𝑛)!) and 𝑏_𝑛=(2^2𝑛−1+1)/(2𝑛)! Then 𝑎_𝑛/𝑏_𝑛=2𝑛/[(2𝑛+1)(2^2𝑛−1+1)]. Moreover, by a simple calculation, we see that 𝑎_𝑛+1𝑏_𝑛+1−𝑎_𝑛𝑏_𝑛=2+2−18𝑛−12𝑛²2^2𝑛−1(2𝑛+1)(2𝑛+3)2^2𝑛−1+12^2𝑛+1+1<0 (3.3) for 𝑛≥1.
Equations (3.1) and (3.2) together with inequality (3.3) and Lemma 2.1 lead to the conclusion that 𝜑(𝑡) is strictly decreasing in (0,log(1+√2)). This in turn implies that lim_𝑡→0+𝜑(𝑡)=29,lim_{√𝑡→log(1+2)}𝜑(𝑡)1=1−√2log1+√2. (3.4)
Therefore, Theorem 1.1 follows from (3.1) and (3.4) together with the monotonicity of 𝜑(𝑡).

Proof of Theorem 1.2. Since 𝐺(𝑎,𝑏), 𝑀(𝑎,𝑏) and 𝑄(𝑎,𝑏) are symmetric and homogeneous of degree 1. Hence, without loss of generality, we assume that 𝑎>𝑏. Let 𝑥=(𝑎−𝑏)/(𝑎+𝑏), 𝑝∈(0,1) and 𝜆₀=1−1/[√2log(1+√2)]. Then making use of 𝐺(𝑎,𝑏)/𝐴(𝑎,𝑏)=√1−𝑥² gives 𝑄(𝑎,𝑏)−𝑀(𝑎,𝑏)𝑄()()=√𝑎,𝑏−𝐺𝑎,𝑏1+𝑥²sinh⁻¹(𝑥)−𝑥√1+𝑥²−√1−𝑥²sinh⁻¹(𝑥). (3.5) Moreover, we obtain lim_𝑥→0+√1+𝑥²sinh⁻¹(𝑥)−𝑥√1+𝑥²−√1−𝑥²sinh⁻¹(𝑥)=13,lim_𝑥→1−√1+𝑥²sinh⁻¹(𝑥)−𝑥√1+𝑥²−√1−𝑥²sinh⁻¹(𝑥)1=1−√2log1+√2=𝜆₀. (3.6)(3.7)
We take the difference between the additive convex combination of 𝐺(𝑎,𝑏), 𝑄(𝑎,𝑏), and 𝑀(𝑎,𝑏) as follows: ()+𝑄()()(𝑝√𝑝𝐺𝑎,𝑏1−𝑝𝑎,𝑏−𝑀𝑎,𝑏=𝐴𝑎,𝑏)1−𝑥²+1−𝑝√1+𝑥²−𝑥sinh⁻¹(𝑥)=𝐴(𝑎,𝑏)𝑝√1−𝑥²+1−𝑝√1+𝑥²sinh⁻¹(𝑥)𝑓_𝑝(𝑥), (3.8) where 𝑓_𝑝(𝑥) is defined as in Lemma 2.2.
Therefore, (1/3)𝐺(𝑎,𝑏)+(2/3)𝑄(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝜆₀𝐺(𝑎,𝑏)+(1−𝜆₀)𝑄(𝑎,𝑏) for all 𝑎,𝑏>0 with 𝑎≠𝑏 follows from (3.8) and Lemma 2.2. This conjunction with the following statement gives the asserted result. (i)If 𝑝<1/3, then (3.5) and (3.6) imply that there exists 0<𝛿₁<1 such that 𝑀(𝑎,𝑏)<𝑝𝐺(𝑎,𝑏)+(1−𝑝)𝑄(𝑎,𝑏) for all 𝑎,𝑏>0 with (𝑎−𝑏)/(𝑎+𝑏)∈(0,𝛿₁).(ii)If 𝑝>𝜆₀, then (3.5) and (3.7) imply that there exists 0<𝛿₂<1 such that 𝑀(𝑎,𝑏)>𝑝𝐺(𝑎,𝑏)+(1−𝑝)𝑄(𝑎,𝑏) for all 𝑎,𝑏>0 with (𝑎−𝑏)/(𝑎+𝑏)∈(1−𝛿₂,1).

Proof of Theorem 1.3. We will follow, to some extent, lines in the proof of Theorem 1.1. First we rearrange terms of (1.9) to obtain 𝛽₃<𝐶(𝑎,𝑏)−𝑀(𝑎,𝑏)𝐶(𝑎,𝑏)−𝐻(𝑎,𝑏)<𝛼₃. (3.9) Use of 𝐶(𝑎,𝑏)/𝐴(𝑎,𝑏)=1+𝑥² followed by a substitution 𝑥=sinh(𝑡) gives 𝛽₃<𝜙(𝑡)<𝛼₃, (3.10) where 𝜙(𝑡)=𝑡[cosh(2𝑡)+1]−2sinh(𝑡)[()],|𝑡|√2𝑡cosh2𝑡−1<log1+2. (3.11)
Since the function 𝜙(𝑡) is an even function, it suffices to investigate its behavior on the interval (0,log(1+√2)).
Using power series of sinh(𝑡) and cosh(𝑡), then (3.11) can be rewritten as 𝜙(𝑡)=∑^∞_𝑛=12^2𝑛/(2𝑛)!−2/(2𝑛+1)!𝑡^2𝑛+1∑^∞_𝑛=12^2𝑛+1/(2𝑛)!𝑡^2𝑛+1. (3.12)
Let 𝑐_𝑛=2^2𝑛/(2𝑛)!−2/(2𝑛+1)! and 𝑑_𝑛=2^2𝑛+1/(2𝑛)!. Then 𝑐_𝑛𝑑_𝑛=12−1(2𝑛+1)2^2𝑛. (3.13) It follows from (3.13) that the sequence {𝑐_𝑛/𝑑_𝑛} is strictly increasing for 𝑛≥1.
Equations (3.12) and (3.13) together with Lemma 2.1 and the monotonicity of {𝑐_𝑛/𝑑_𝑛} lead to the conclusion that 𝜙(𝑡) is strictly increasing in (0,log(1+√2)). Moreover, lim_𝑡→0+𝜙(𝑡)=𝑐₁𝑑₁=512,lim_{√𝑡→log(1+2)}𝜙(𝑡)1=1−2log1+√2. (3.14)
Making use of (3.14) and (3.10) together with the monotonicity of 𝜙(𝑡) gives the asserted result.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11271105, and Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.