Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means

Abstract

We give several sharp bounds for the Neuman means 𝑁_𝐴𝐻 and 𝑁_𝐻𝐴 (𝑁_𝐶𝐴 and 𝑁_𝐴𝐶) in terms of harmonic mean H (contraharmonic mean C) or the geometric convex combination of arithmetic mean A and harmonic mean H (contraharmonic mean C and arithmetic mean A) and present a new chain of inequalities for certain bivariate means.

1. Introduction

For 𝑎,𝑏>0 with 𝑎≠𝑏, the Schwab-Borchardt mean SB(𝑎,𝑏) [1–3] of 𝑎 and 𝑏 is defined as SB(𝑎,𝑏)=𝑏²−𝑎²cos⁻¹(𝑎/𝑏),𝑎<𝑏,𝑎²−𝑏²cosh⁻¹(𝑎/𝑏),𝑎>𝑏, (1) where cos⁻¹(𝑥) and cosh⁻¹(𝑥)=log(𝑥+√𝑥²−1) are the inverse cosine and inverse hyperbolic cosine functions, respectively.

The Schwab-Borchardt mean SB(𝑎,𝑏) can be expressed by the symmetric elliptic integral 𝑅_𝐹 [4] of the first kind as follows [5] (see also [6, (3.21)]): ()=1SB𝑎,𝑏𝑅_𝐹𝑎²,𝑏²,𝑏², (2) where 𝑅_𝐹(𝑎,𝑏,𝑐)=(1/2)∫^∞₀[(𝑡+𝑎)(𝑡+𝑏)(𝑡+𝑐)]^−1/2𝑑𝑡.

Recently, the Schwab-Borchardt mean has been the subject of intensive research. In particular, many remarkable inequalities for the Schwab-Borchardt mean and its generated means can be found in the literature [1–3, 7–10].

Very recently, Neuman [11] found a new mean 𝑁(𝑎,𝑏) derived from the Schwab-Borchardt mean as follows: 𝑁()=1𝑎,𝑏2𝑏𝑎+²().SB𝑎,𝑏 (3)

Let 𝐺(𝑎,𝑏)=√𝑎𝑏, 𝐻(𝑎,𝑏)=2𝑎𝑏/(𝑎+𝑏), 𝐿(𝑎,𝑏)=(𝑎−𝑏)/(log𝑎−log𝑏), 𝑃(𝑎,𝑏)=(𝑎−𝑏)/[2sin⁻¹((𝑎−𝑏)/(𝑎+𝑏))], 𝐴(𝑎,𝑏)=(𝑎+𝑏)/2, 𝑀(𝑎,𝑏)=(𝑎−𝑏)/[2sinh⁻¹((𝑎−𝑏)/(𝑎+𝑏))], 𝑇(𝑎,𝑏)=(𝑎−𝑏)/[2tan⁻¹((𝑎−𝑏)/(𝑎+𝑏))], 𝑄(𝑎,𝑏)=(𝑎²+𝑏²)/2, and 𝐶(𝑎,𝑏)=(𝑎²+𝑏²)/(𝑎+𝑏) be, respectively, the geometric, harmonic, logarithmic, first Seiffert, arithmetic, Neuman-Sándor, second Seiffert, quadratic, and contraharmonic means, and let 𝑁_𝐴𝐻()[𝐴()(,𝑁𝑎,𝑏=𝑁𝑎,𝑏,𝐻𝑎,𝑏)]_𝐻𝐴()[𝐻()(,𝑁𝑎,𝑏=𝑁𝑎,𝑏,𝐴𝑎,𝑏)]_𝐴𝐺()[𝐴()(,𝑁𝑎,𝑏=𝑁𝑎,𝑏,𝐺𝑎,𝑏)]_𝐺𝐴()[𝐺()(,𝑁𝑎,𝑏=𝑁𝑎,𝑏,𝐴𝑎,𝑏)]_𝐴𝐶()[𝐴()(,𝑁𝑎,𝑏=𝑁𝑎,𝑏,𝐶𝑎,𝑏)]_𝐶𝐴()[𝐶()(,𝑁𝑎,𝑏=𝑁𝑎,𝑏,𝐴𝑎,𝑏)]_𝐴𝑄()[𝐴()(,𝑁𝑎,𝑏=𝑁𝑎,𝑏,𝑄𝑎,𝑏)]_𝑄𝐴(𝑎,𝑏)=𝑁[𝑄(𝑎,𝑏),𝐴(𝑎,𝑏)] (4) be the Neuman means. Then Neuman [11] proved that 𝐺(𝑎,𝑏)<𝐿(𝑎,𝑏)<𝑁_𝐴𝐺(𝑎,𝑏)<𝑃(𝑎,𝑏)<𝑁_𝐺𝐴()()()𝑎,𝑏<𝐴𝑎,𝑏<𝑀𝑎,𝑏<𝑁_𝑄𝐴(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝑁_𝐴𝑄(𝑎,𝑏)<𝑄(𝑎,𝑏)<𝐶(𝑎,𝑏), (5) for all 𝑎,𝑏>0 with 𝑎≠𝑏, and the double inequalities 𝛼₁𝐴(𝑎,𝑏)+1−𝛼₁𝐺(𝑎,𝑏)<𝑁_𝐺𝐴(𝑎,𝑏)<𝛽₁𝐴(𝑎,𝑏)+1−𝛽₁𝐺(),𝛼𝑎,𝑏₂𝑄(𝑎,𝑏)+1−𝛼₂𝐴(𝑎,𝑏)<𝑁_𝐴𝑄(𝑎,𝑏)<𝛽₂𝑄(𝑎,𝑏)+1−𝛽₂𝐴(),𝛼𝑎,𝑏₃𝐴(𝑎,𝑏)+1−𝛼₃𝐺(𝑎,𝑏)<𝑁_𝐴𝐺(𝑎,𝑏)<𝛽₃𝐴(𝑎,𝑏)+1−𝛽₃𝐺(),𝛼𝑎,𝑏₄𝑄(𝑎,𝑏)+1−𝛼₄𝐴(𝑎,𝑏)<𝑁_𝑄𝐴(𝑎,𝑏)<𝛽₄𝑄(𝑎,𝑏)+1−𝛽₄𝐴(𝑎,𝑏) (6) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼₁≤2/3, 𝛽₁≥𝜋/4, 𝛼₂≤2/3, 𝛽₂≥(𝜋−2)/[4(√2−1)]=0.689…, 𝛼₃≤1/3, 𝛽₃≥1/2, 𝛼₄≤1/3, and 𝛽₄≥[log(1+√2)+√2−2]/[2(√2−1)]=0.356….

Zhang et al. [12] presented the best possible parameters 𝛼₁,𝛼₂,𝛽₁,𝛽₂∈[0,1/2] and 𝛼₃,𝛼₄,𝛽₃,𝛽₄∈[1/2,1] such that the double inequalities 𝐺𝛼₁𝑎+1−𝛼₁𝑏,𝛼₁𝑏+1−𝛼₁𝑎<𝑁_𝐴𝐺(𝑎,𝑏)<𝐺𝛽₁𝑎+1−𝛽₁𝑏,𝛽₁𝑏+1−𝛽₁𝑎,𝐺𝛼₂𝑎+1−𝛼₂𝑏,𝛼₂𝑏+1−𝛼₂𝑎<𝑁_𝐺𝐴(𝑎,𝑏)<𝐺𝛽₂𝑎+1−𝛽₂𝑏,𝛽₂𝑏+1−𝛽₂𝑎,𝑄𝛼₃𝑎+1−𝛼₃𝑏,𝛼₃𝑏+1−𝛼₃𝑎<𝑁_𝑄𝐴(𝑎,𝑏)<𝑄𝛽₃𝑎+1−𝛽₃𝑏,𝛽₃𝑏+1−𝛽₃𝑎,𝑄𝛼₄𝑎+1−𝛼₄𝑏,𝛼₄𝑏+1−𝛼₄𝑎<𝑁_𝐴𝑄(𝑎,𝑏)<𝑄𝛽₄𝑎+1−𝛽₄𝑏,𝛽₄𝑏+1−𝛽₄𝑎 (7) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏.

In [13], the authors found the greatest values 𝛼₁, 𝛼₂, 𝛼₃, 𝛼₄, 𝛼₅, 𝛼₆, 𝛼₇, and 𝛼₈ and the least values 𝛽₁, 𝛽₂, 𝛽₃, 𝛽₄, 𝛽₅, 𝛽₆, 𝛽₇, and 𝛽₈ such that the double inequalities 𝐴^𝛼1(𝑎,𝑏)𝐺^1−𝛼1(𝑎,𝑏)<𝑁_𝐺𝐴(𝑎,𝑏)<𝐴^𝛽1(𝑎,𝑏)𝐺^1−𝛽1(),𝛼𝑎,𝑏₂𝐺()+𝑎,𝑏1−𝛼₂𝐴()<1𝑎,𝑏𝑁_𝐺𝐴()<𝛽𝑎,𝑏𝐺(²)+𝑎,𝑏1−𝛽₂𝐴(),𝐴𝑎,𝑏_𝛼3(𝑎,𝑏)𝐺^1−𝛼3(𝑎,𝑏)<𝑁_𝐴𝐺(𝑎,𝑏)<𝐴^𝛽3(𝑎,𝑏)𝐺^1−𝛽3(),𝛼𝑎,𝑏₄𝐺()+𝑎,𝑏1−𝛼₄𝐴()<1𝑎,𝑏𝑁_𝐴𝐺()<𝛽𝑎,𝑏𝐺(⁴)+𝑎,𝑏1−𝛽₄𝐴(),𝑄𝑎,𝑏_𝛼5(𝑎,𝑏)𝐴^1−𝛼5(𝑎,𝑏)<𝑁_𝐴𝑄(𝑎,𝑏)<𝑄^𝛽5(𝑎,𝑏)𝐴^1−𝛽5(),𝛼𝑎,𝑏₆𝐴()+𝑎,𝑏1−𝛼₆𝑄()<1𝑎,𝑏𝑁_𝐴𝑄()<𝛽𝑎,𝑏𝐴(⁶)+𝑎,𝑏1−𝛽₆𝑄(),𝑄𝑎,𝑏_𝛼7(𝑎,𝑏)𝐴^1−𝛼7(𝑎,𝑏)<𝑁_𝑄𝐴(𝑎,𝑏)<𝑄^𝛽7(𝑎,𝑏)𝐴^1−𝛽7(),𝛼𝑎,𝑏₈𝐴()+𝑎,𝑏1−𝛼₈𝑄()<1𝑎,𝑏𝑁_𝑄𝐴()<𝛽𝑎,𝑏𝐴(⁸)+𝑎,𝑏1−𝛽₈𝑄(𝑎,𝑏) (8) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏.

Let 𝑎>𝑏>0, 𝑣=(𝑎−𝑏)/(𝑎+𝑏)∈(0,1), 𝑝∈(0,∞), 𝑞∈(0,𝜋/2), 𝑟∈(0,log(2+√3)), and 𝑠∈(0,𝜋/3) be the parameters such that 1/cosh(𝑝)=cos(𝑞)=1−𝑣², cosh(𝑟)=sec(𝑠)=1+𝑣². Then He et al. [14] proved that 𝑁_𝐴𝐻()=1𝑎,𝑏2𝐴(𝑎,𝑏)1+2𝑝,sinh2𝑝𝑁_𝐻𝐴()=1𝑎,𝑏2𝐴(𝑞+𝑞𝑎,𝑏)cos𝑞,sin𝑁_𝐶𝐴()=1𝑎,𝑏2𝐴((𝑟)+𝑟𝑎,𝑏)cosh(𝑟),sinh𝑁_𝐴𝐶()=1𝑎,𝑏2𝐴(𝑎,𝑏)1+2𝑠(),sin2𝑠𝐻(𝑎,𝑏)<𝑁_𝐴𝐻(𝑎,𝑏)<𝑁_𝐻𝐴(𝑎,𝑏)<𝐴(𝑎,𝑏)<𝑁_𝐶𝐴(𝑎,𝑏)<𝑁_𝐴𝐶(𝑎,𝑏)<𝐶(𝑎,𝑏). (9)(10)(11)(12)(13)

Let 𝑥∈[0,1/2], 𝑦∈[1/2,1], 𝑓(𝑥)=𝐻[𝑥𝑎+(1−𝑥)𝑏,𝑥𝑏+(1−𝑥)𝑎], and 𝑔(𝑦)=𝐶[𝑦𝑎+(1−𝑦)𝑏,𝑦𝑏+(1−𝑦)𝑎]. Then we clearly see that 𝑓(0)=𝐻(𝑎,𝑏)<𝑁_𝐴𝐻(𝑎,𝑏)<𝑁_𝐻𝐴()()1𝑎,𝑏<𝐴𝑎,𝑏=𝑓21=𝑔2<𝑁_𝐶𝐴(𝑎,𝑏)<𝑁_𝐴𝐶()()(1),𝑎,𝑏<𝐶𝑎,𝑏=𝑔 (14) for all 𝑎,𝑏>0 with 𝑎≠𝑏, and the functions 𝑓 and 𝑔 are, respectively, strictly increasing on the intervals [0,1/2] and [1/2,1] for fixed 𝑎,𝑏>0 with 𝑎≠𝑏.

The main purpose of this paper is to find the best possible parameters 𝛼₁, 𝛼₂, 𝛼₃, 𝛼₄, 𝛽₁, 𝛽₂, 𝛽₃, 𝛽₄∈ℝ, 𝛼₅,𝛼₆,𝛽₅,𝛽₆∈[0,1/2], and 𝛼₇,𝛼₈,𝛽₇,𝛽₈∈[1/2,1] such that the double inequalities 𝐴^𝛼1(𝑎,𝑏)𝐻^1−𝛼1(𝑎,𝑏)<𝑁_𝐴𝐻(𝑎,𝑏)<𝐴^𝛽1(𝑎,𝑏)𝐻^1−𝛽1(),𝐴𝑎,𝑏_𝛼2(𝑎,𝑏)𝐻^1−𝛼2(𝑎,𝑏)<𝑁_𝐻𝐴(𝑎,𝑏)<𝐴^𝛽2(𝑎,𝑏)𝐻^1−𝛽2(),𝐶𝑎,𝑏_𝛼3(𝑎,𝑏)𝐴^1−𝛼3(𝑎,𝑏)<𝑁_𝐶𝐴(𝑎,𝑏)<𝐶^𝛽3(𝑎,𝑏)𝐴^1−𝛽3(),𝐶𝑎,𝑏_𝛼4(𝑎,𝑏)𝐴^1−𝛼4(𝑎,𝑏)<𝑁_𝐴𝐶(𝑎,𝑏)<𝐶^𝛽4(𝑎,𝑏)𝐴^1−𝛽4(),𝐻𝛼𝑎,𝑏₅𝑎+1−𝛼₅𝑏,𝛼₅𝑏+1−𝛼₅𝑎<𝑁_𝐴𝐻(𝑎,𝑏)<𝐻𝛽₅𝑎+1−𝛽₅𝑏,𝛽₅𝑏+1−𝛽₅𝑎,𝐻𝛼₆𝑎+1−𝛼₆𝑏,𝛼₆𝑏+1−𝛼₆𝑎<𝑁_𝐻𝐴(𝑎,𝑏)<𝐻𝛽₆𝑎+1−𝛽₆𝑏,𝛽₆𝑏+1−𝛽₆𝑎,𝐶𝛼₇𝑎+1−𝛼₇𝑏,𝛼₇𝑏+1−𝛼₇𝑎<𝑁_𝐶𝐴(𝑎,𝑏)<𝐻𝛽₇𝑎+1−𝛽₇𝑏,𝛽₇𝑏+1−𝛽₇𝑎,𝐶𝛼₈𝑎+1−𝛼₈𝑏,𝛼₈𝑏+1−𝛼₈𝑎<𝑁_𝐴𝐶(𝑎,𝑏)<𝐻𝛽₈𝑎+1−𝛽₈𝑏,𝛽₈𝑏+1−𝛽₈𝑎 (15) hold, for all 𝑎,𝑏>0 with 𝑎≠𝑏, and present a new chain of inequalities for certain bivariate means.

2. Lemmas

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

Lemma 1 (see [15, Theorem 1.25]). For −∞<𝑎<𝑏<∞, let 𝑓,𝑔∶[𝑎,𝑏]→ℝ be continuous on [𝑎,𝑏] and differentiable on (𝑎,𝑏); let 𝑔^′(𝑥)≠0 on (𝑎,𝑏). If 𝑓^′(𝑥)/𝑔^′(𝑥) is increasing (decreasing) on (𝑎,𝑏), then so are 𝑓(𝑥)−𝑓(𝑎)𝑔(𝑥)(𝑎),𝑓(𝑥)(𝑏)−𝑔−𝑓𝑔(𝑥)(𝑏).−𝑔 (16) If 𝑓^′(𝑥)/𝑔^′(𝑥) is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2 (see [16, 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 𝑎_𝑛,𝑏_𝑛>0 for all 𝑛≥0. If the sequence {𝑎_𝑛/𝑏_𝑛} is (strictly) increasing (decreasing) for all 𝑛≥0, then the function 𝑓(𝑥)/𝑔(𝑥) is also (strictly) increasing (decreasing) on (0,𝑟).

Lemma 3. The function 𝜑₁(𝑥)=sinh(𝑥)−𝑥sinh(𝑥)[cosh(𝑥)−1] (17) is strictly decreasing from (0,∞) onto (0,1/3).

Proof. Making use of power series expansion we get 𝜑₁(𝑥)=2sinh(𝑥)−2𝑥()(𝑥)=∑sinh2𝑥−2sinh_∞𝑛=1(2/(2𝑛+1)!)𝑥^2𝑛+1∑^∞_𝑛=12^2𝑛+1−2/(2𝑛+1)!𝑥^2𝑛+1=∑^∞_𝑛=0(2/(2𝑛+3)!)𝑥^2𝑛∑^∞_𝑛=02^2𝑛+3−2/(2𝑛+3)!𝑥^2𝑛. (18)
Let 𝑎_𝑛=2(2𝑛+3)!,𝑏_𝑛=2^2𝑛+3−2()!.2𝑛+3 (19) Then 𝑎_𝑛>0,𝑏_𝑛>0, (20) and 𝑎_𝑛/𝑏_𝑛=1/(2^2𝑛+2−1) is strictly decreasing for all 𝑛≥0.
Note that 𝜑₁0⁺=𝑎₀𝑏₀=13,𝜑₁(∞)=lim_𝑛→∞𝑎_𝑛𝑏_𝑛=0. (21)
Therefore, Lemma 3 follows easily from Lemma 2 and (18)–(21) together with the monotonicity of the sequence {𝑎_𝑛/𝑏_𝑛}.

Lemma 4. The function 𝜑₂(𝑥)=𝑥−sin(𝑥)sin(𝑥)[1−cos(𝑥)] (22) is strictly increasing from (0,𝜋/2) onto (1/3,𝜋/2−1).

Proof. Let 𝑓₁(𝑥)=𝑥−sin(𝑥) and 𝑔₁(𝑥)=sin(𝑥)[1−cos(𝑥)]. Then simple computations lead to 𝜑₂(𝑥)=𝑓₁(𝑥)𝑔₁(𝑥)=𝑓₁(𝑥)−𝑓₁(0)𝑔₁(𝑥)−𝑔₁(0), (23) and 𝑓^′₁(𝑥)/𝑔^′₁(𝑥)=1/[1+2cos(𝑥)] is strictly increasing on (0,𝜋/2).
Note that 𝜑₂0⁺=lim_𝑥→0+𝑓^′₁(𝑥)𝑔^′₁(𝑥)=13,𝜑₂𝜋2=𝜋2−1. (24)
Therefore, Lemma 4 follows from Lemma 1, (23), (24), and the monotonicity of 𝑓^′₁(𝑥)/𝑔^′₁(𝑥).

Lemma 5. The function 𝜑₃(𝑥)=log[(𝑥+sinh(𝑥)cosh(𝑥))/(2sinh(𝑥)cosh(𝑥))]log(1/cosh(𝑥)) (25) is strictly decreasing from (0,∞) onto (0,2/3).

Proof. Let 𝑓₂(𝑥)=log[(𝑥+sinh(𝑥)cosh(𝑥))/(2sinh(𝑥)cosh(𝑥))] and 𝑔₂(𝑥)=log[1/cosh(𝑥)]. Then simple computations lead to 𝜑₃(𝑥)=𝑓₂(𝑥)𝑔₂(𝑥)=𝑓₂(𝑥)−𝑓₂0⁺𝑔₂(𝑥)−𝑔₂(0),𝑓^′₂(𝑥)𝑔^′₂(𝑥)=𝑥cosh³(𝑥)+sinh²(𝑥)cosh(𝑥)−sinh(𝑥)cosh²(𝑥)×sinh²(𝑥)cosh(𝑥)[𝑥+sinh(𝑥)cosh(𝑥)]⁻¹=(()(𝑥)()(𝑥×(()(𝑥)()())8𝑥cosh3𝑥+8𝑥cosh−4sinh3𝑥−4sinh))4𝑥cosh3𝑥−4𝑥cosh+sinh5𝑥−sinh3𝑥−2sinh(𝑥)⁻¹=^∞_𝑛=18(2𝑛+1)1+3^2𝑛−41+3^2𝑛+1×((2𝑛+1)!)⁻¹𝑥^2𝑛+1×^∞_𝑛=14(2𝑛+1)3^2𝑛−1+5^2𝑛+1−3^2𝑛+1×()−2(2𝑛+1)!⁻¹𝑥^2𝑛+1−1=^∞_𝑛=08(2𝑛+3)1+3^2𝑛+2−41+3^2𝑛+3×((2𝑛+3)!)⁻¹𝑥^2𝑛×^∞_𝑛=04(2𝑛+3)3^2𝑛+2−1+5^2𝑛+3−3^2𝑛+3×()−2(2𝑛+3)!⁻¹𝑥^2𝑛−1. (26)(27)
Let𝑎_𝑛=8(2𝑛+3)1+3^2𝑛+2−41+3^2𝑛+3()!,𝑏2𝑛+3_𝑛=4(2𝑛+3)3^2𝑛+2−1+5^2𝑛+3−3^2𝑛+3−2()!.2𝑛+3 (28) Then it is not difficult to verify that 𝑎_𝑛=(16𝑛+20)+(16𝑛+12)3^2𝑛+2(2𝑛+3)!>0,𝑏_𝑛𝑎>0,_𝑛+1𝑏_𝑛+1−𝑎_𝑛𝑏_𝑛𝑐=−^𝑛𝑑_𝑛<0 (29) for all 𝑛≥0, where 𝑐_𝑛=(256𝑛+48)3^2𝑛+2×5^2𝑛+3+(384𝑛+464)5^2𝑛+3+(2048𝑛²+7040𝑛+6000)3^2𝑛+2−16⋅3^4𝑛+7+64 and 𝑑_𝑛=[5^2𝑛+5+(8𝑛+17)3^2𝑛+4−(8𝑛+22)][5^2𝑛+3+(8𝑛+9)3^2𝑛+2−(8𝑛+14)].
Note that 𝜑₃0⁺=𝑎₀𝑏₀=23,𝜑₃(∞)=0. (30)
It follows from Lemma 2 and (27)–(29) that the function 𝑓^′₂(𝑥)/𝑔^′₂(𝑥) is strictly decreasing on (0,∞). Therefore, Lemma 5 follows from (26) and (30) together with Lemma 1 and the monotonicity of 𝑓^′₂(𝑥)/𝑔^′₂(𝑥).

Lemma 6. The function 𝜑₄(𝑥)=log[(𝑥+sin(𝑥)cos(𝑥))/(2sin(𝑥))]log[cos(𝑥)] (31) is strictly decreasing from (0,𝜋/2) onto (0,1/3).

Proof. Let 𝑓₃(𝑥)=log[(𝑥+sin(𝑥)cos(𝑥))/(2sin(𝑥))], 𝑔₃(𝑥)=log[cos(𝑥)], 𝑓₄(𝑥)=cos²(𝑥)[𝑥−sin(𝑥)cos(𝑥)], and 𝑔₄(𝑥)=sin²(𝑥)[𝑥+sin(𝑥)cos(𝑥)]. Then simple computations lead to 𝜑₄(𝑥)=𝑓₃(𝑥)𝑔₃(𝑥)=𝑓₃(𝑥)−𝑓₃0⁺𝑔₃(𝑥)−𝑔₃(0),𝑓^′₃(𝑥)𝑔^′₃(𝑥)=𝑓₄(𝑥)𝑔₄(𝑥)=𝑓₄(𝑥)−𝑓₄(0)𝑔₄(𝑥)−𝑔₄(0),𝑓^′₄(𝑥)𝑔^′₄(𝑥)1=1−()/().1/2+sin2𝑥2𝑥 (32)(33)(34)
Since the function 𝑥→sin(𝑥)/𝑥 is strictly decreasing on (0,𝜋), hence (34) leads to the conclusion that 𝑓^′₄(𝑥)/𝑔^′₄(𝑥) is strictly decreasing on (0,𝜋/2).
Note that 𝜑₄0⁺=lim_𝑥→0+𝑓^′₄(𝑥)𝑔^′₄(𝑥)=13,𝜑₄(∞)=0. (35)
Therefore, Lemma 6 follows easily from (32), (33), (35), and Lemma 1 together with the monotonicity of 𝑓^′₄(𝑥)/𝑔^′₄(𝑥).

Lemma 7. The function 𝜑₅(𝑥)=log((𝑥+sinh(𝑥)cosh(𝑥))/(2sinh(𝑥)))log[cosh(𝑥)] (36) is strictly increasing from (0,log(2+√3)) onto (1/3,[2log(2√3+log(2+√3))−log3]/(2log2)−1).

Proof. Let 𝑓₅(𝑥)=log[(𝑥+sinh(𝑥)cosh(𝑥))/(2sinh(𝑥))] and 𝑔₅(𝑥)=log[cosh(𝑥)]. Then simple computations lead to 𝜑₅(𝑥)=𝑓₅(𝑥)𝑔₅(𝑥)=𝑓₅(𝑥)−𝑓₅0⁺𝑔₅(𝑥)−𝑔₅(0),𝑓^′₅(𝑥)𝑔^′₅(𝑥)=sinh(𝑥)cosh³(𝑥)−𝑥cosh²(𝑥)𝑥sinh²(𝑥)+sinh³(𝑥)(𝑥)=()()()coshsinh4𝑥+2sinh2𝑥−4𝑥cosh2𝑥−4𝑥()()()=∑sinh4𝑥−2sinh2𝑥+4𝑥cosh2𝑥−4𝑥_∞𝑛=12^2𝑛+22^2𝑛−2𝑛/(2𝑛+1)!𝑥^2𝑛+1∑^∞_𝑛=12^2𝑛+22^2𝑛+2𝑛/(2𝑛+1)!𝑥^2𝑛+1=∑^∞_𝑛=02^2𝑛+42^2𝑛+2−(2𝑛+2)/(2𝑛+3)!𝑥^2𝑛∑^∞_𝑛=02^2𝑛+42^2𝑛+2+(2𝑛+2)/(2𝑛+3)!𝑥^2𝑛. (37)(38)
Let 𝑎_𝑛=2^2𝑛+42^2𝑛+2−(2𝑛+2)()!,𝑏2𝑛+3_𝑛=2^2𝑛+42^2𝑛+2+(2𝑛+2)()!.2𝑛+3 (39)
Then 𝑎_𝑛>0,𝑏_𝑛𝑎>0,_𝑛+1𝑏_𝑛+1−𝑎_𝑛𝑏_𝑛=(3𝑛+2)2^2𝑛+22^2𝑛+1+𝑛+12^2𝑛+3+𝑛+2>0 (40) for all 𝑛≥0.
It follows from Lemma 2 and (38)–(40) that 𝑓^′₅(𝑥)/𝑔^′₅(𝑥) is strictly increasing on (0,log(2+√3)).
Note that 𝜑₅0⁺=𝑎₀𝑏₀=13𝜑₅log2+√3=2log2√3+log2+√3−log32log2−1. (41)
Therefore, Lemma 7 follows from Lemma 1, (37), and (41) together with the monotonicity of 𝑓^′₅(𝑥)/𝑔^′₅(𝑥).

Lemma 8. The function 𝜑₆(𝑥)=log[(𝑥+sin(𝑥)cos(𝑥))/(2sin(𝑥)cos(𝑥))]log[sec(𝑥)] (42) is strictly increasing from (0,𝜋/3) onto (2/3,[2log(4𝜋+3√3)−2log6−log3]/(2log2)).

Proof. Let 𝑓₆(𝑥)=log[(𝑥+sin(𝑥)cos(𝑥))/(2sin(𝑥)cos(𝑥))], 𝑔₆(𝑥)=log[sec(𝑥)], 𝑓₇(𝑥)=sin(𝑥)cos(𝑥)−𝑥cos(2𝑥), and 𝑔₇(𝑥)=(𝑥+sin(𝑥)cos(𝑥))sin²(𝑥). Then simple computations lead to 𝜑₆(𝑥)=𝑓₆(𝑥)𝑔₆(𝑥)=𝑓₆(𝑥)−𝑓₆0⁺𝑔₆(𝑥)−𝑔₆(0),𝑓^′₆(𝑥)𝑔^′₆(𝑥)=𝑓₇(𝑥)𝑔₇(𝑥)=𝑓₇(𝑥)−𝑓₇(0)𝑔₇(𝑥)−𝑔₇(0),𝑓^′₇(𝑥)𝑔^′₇(𝑥)=1().1/2+sin2𝑥/2𝑥 (43)
Since the function 𝑥→sin(2𝑥)/(2𝑥) is strictly decreasing on (0,𝜋/3), hence Lemma 1 and (43) lead to the conclusion that 𝜑₆(𝑥) is strictly increasing on (0,𝜋/3).
Note that 𝜑₆0⁺=lim_𝑥→0+𝑓^′₇(𝑥)𝑔^′₇(𝑥)=23,𝜑₆𝜋3=2log4𝜋+3√3−2log6−log3.2log2 (44)(45)
Therefore, Lemma 8 follows easily from (44) and (45) together with the monotonicity of 𝜑₆(𝑥).

3. Main Results

Theorem 9. The double inequalities 𝐴^𝛼1(𝑎,𝑏)𝐻^1−𝛼1(𝑎,𝑏)<𝑁_𝐴𝐻(𝑎,𝑏)<𝐴^𝛽1(𝑎,𝑏)𝐻^1−𝛽1(𝑎,𝑏),𝐴^𝛼2(𝑎,𝑏)𝐻^1−𝛼2(𝑎,𝑏)<𝑁_𝐻𝐴(𝑎,𝑏)<𝐴^𝛽2(𝑎,𝑏)𝐻^1−𝛽2(𝑎,𝑏),𝐶^𝛼3(𝑎,𝑏)𝐴^1−𝛼3(𝑎,𝑏)<𝑁_𝐶𝐴(𝑎,𝑏)<𝐶^𝛽3(𝑎,𝑏)𝐴^1−𝛽3(𝑎,𝑏),𝐶^𝛼4(𝑎,𝑏)𝐴^1−𝛼4(𝑎,𝑏)<𝑁_𝐴𝐶(𝑎,𝑏)<𝐶^𝛽4(𝑎,𝑏)𝐴^1−𝛽4(𝑎,𝑏) (46)(47)(48)(49) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼₁≤1/3, 𝛽₁≥1, 𝛼₂≤2/3, 𝛽₂≥1, 𝛼₃≤1/3, 𝛽₃≥[2log(2√3+log(2+√3))−log3]/(2log2)−1=0.4648…, 𝛼₄≤2/3, and 𝛽₄≥[2log(4𝜋+3√3)−2log6−log3]/(2log2)=0.7733….

Proof. Without loss of generality, we assume that 𝑎>𝑏>0. Let 𝑣=(𝑎−𝑏)/(𝑎+𝑏)∈(0,1), 𝑝∈(0,∞), 𝑞∈(0,𝜋/2), 𝑟∈(0,log(2+√3)), and 𝑠∈(0,𝜋/3) be the parameters such that 1/cosh(𝑝)=cos(𝑞)=1−𝑣², cosh(𝑟)=sec(𝑠)=1+𝑣². Then (9)–(12) lead to the conclusion that inequalities (46)–(49) are, respectively, equivalent to 𝛼₁<1−log𝑁_𝐴𝐻(𝑎,𝑏)−log𝐴(𝑎,𝑏)()()𝑝𝑝/𝑝𝑝log𝐻𝑎,𝑏−log𝐴𝑎,𝑏=1−log𝑝+sinhcosh2sinhcoshlog1/cosh𝑝=1−𝜑₃𝑝<𝛽₁,𝛼₂<1−log𝑁_𝐻𝐴(𝑎,𝑏)−log𝐴(𝑎,𝑏)()()𝑞𝑞/𝑞log𝐻𝑎,𝑏−log𝐴𝑎,𝑏=1−log𝑞+sincos2sinlogcos𝑞=1−𝜑₄𝑞<𝛽₂,𝛼₃<log𝑁_𝐶𝐴(𝑎,𝑏)−log𝐴(𝑎,𝑏)()()=(𝑟)(𝑟/((𝑟log𝐶𝑎,𝑏−log𝐴𝑎,𝑏log((𝑟+sinhcosh))2sinh)))log[cosh(𝑟)]=𝜑₅(𝑟)<𝛽₃,𝛼₄<log𝑁_𝐴𝐶(𝑎,𝑏)−log𝐴(𝑎,𝑏)()()=(𝑠)(𝑠/((𝑠)(𝑠log𝐶𝑎,𝑏−log𝐴𝑎,𝑏log[(𝑠+sincos))2sincos))]log[sec(𝑠)]=𝜑₆(𝑠)<𝛽₄. (50)(51)(52)(53)
Therefore, Theorem 9 follows easily from (50)–(53) and Lemmas 5–8.

Theorem 10. Let 𝛼₅,𝛽₅,𝛼₆,𝛽₆∈[0,1/2]. Then the double inequalities, 𝐻𝛼₅𝑎+1−𝛼₅𝑏,𝛼₅𝑏+1−𝛼₅𝑎<𝑁_𝐴𝐻(𝑎,𝑏)<𝐻𝛽₅𝑎+1−𝛽₅𝑏,𝛽₅𝑏+1−𝛽₅𝑎,𝐻𝛼₆𝑎+1−𝛼₆𝑏,𝛼₆𝑏+1−𝛼₆𝑎<𝑁_𝐻𝐴(𝑎,𝑏)<𝐻𝛽₆𝑎+1−𝛽₆𝑏,𝛽₆𝑏+1−𝛽₆𝑎, (54) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼₅≤1/2−√6/6=0.09175…, 𝛽₅≥1/2−√2/4=0.1464…, 𝛼₆≤1/2−√3/6=0.2113…, and 𝛽₆≥1/2−1−𝜋/4/2=0.2683….

Proof. Without loss of generality, we assume that 𝑎>𝑏>0. Let 𝜆∈[0,1/2], 𝑣=(𝑎−𝑏)/(𝑎+𝑏)∈(0,1), 𝑝∈(0,∞), and 𝑞∈(0,𝜋/2) be the parameters such that 1/cosh(𝑝)=cos(𝑞)=1−𝑣². Then from (9) and (10) we have 𝐻[𝜆𝑎+(1−𝜆)𝑏,𝜆𝑏+(1−𝜆)𝑎]−𝑁_𝐴𝐻()(𝑎,𝑏=𝐴𝑎,𝑏)1−(1−2𝜆)²𝑣²−12𝐴(𝑝𝑎,𝑏)1+𝑝𝑝=𝐴()sinhcosh𝑎,𝑏2×1−2(1−2𝜆)²cosh𝑝−1𝑝−𝑝cosh𝑝𝑝=𝐴(𝑝sinhcosh𝑎,𝑏)cosh−1𝑝×𝑝2cosh1+sinh−𝑝sinh𝑝cosh𝑝−1−2(1−2𝜆)²=𝐴(𝑎,𝑏)cosh𝑝−12cosh𝑝1+𝜑₁𝑝−2(1−2𝜆)²,𝐻[𝜆𝑎+(1−𝜆)𝑏,𝜆𝑏+(1−𝜆)𝑎]−𝑁_𝐻𝐴()(𝑎,𝑏=𝐴𝑎,𝑏)1−(1−2𝜆)²𝑣²−12𝐴(𝑞+𝑞𝑎,𝑏)cos𝑞=1sin2𝐴()×𝑎,𝑏2−2(1−2𝜆)²𝑞𝑞−𝑞1−cos−cos𝑞=𝑞sin1−cos2𝐴(𝑎,𝑏)1−2(1−2𝜆)²−𝜑₂𝑞. (55)(56)
Therefore, Theorem 10 follows easily from (55) and (56) together with Lemmas 3 and 4.

Theorem 11. Let 𝛼₇,𝛽₇,𝛼₈,𝛽₈∈[1/2,1]. Then the double inequalities, 𝐶𝛼₇𝑎+1−𝛼₇𝑏,𝛼₇𝑏+1−𝛼₇𝑎<𝑁_𝐶𝐴(𝑎,𝑏)<𝐶𝛽₇𝑎+1−𝛽₇𝑏,𝛽₇𝑏+1−𝛽₇𝑎,𝐶𝛼₈𝑎+1−𝛼₈𝑏,𝛼₈𝑏+1−𝛼₈𝑎<𝑁_𝐴𝐶(𝑎,𝑏)<𝐻𝛽₈𝑎+1−𝛽₈𝑏,𝛽₈𝑏+1−𝛽₈𝑎, (57) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼₇≤1/2+√3/6=0.7886…, 𝛽₇≥1/2+6√3log(2+√3)/12=0.8082…, 𝛼₈≤1/2+√6/6=0.9082…, and 𝛽₆≥1/2+8√3𝜋−18/12=0.9210….

Proof. Without loss of generality, we assume that 𝑎>𝑏>0. Let 𝜇∈[1/2,1], 𝑣=(𝑎−𝑏)/(𝑎+𝑏)∈(0,1), 𝑟∈(0,log(2+√3)), and 𝑠∈(0,𝜋/3) be the parameters such that cosh(𝑟)=sec(𝑠)=1+𝑣². Then from (11) and (12) one has 𝐶𝜇𝑎+1−𝜇𝑏,𝜇𝑏+1−𝜇𝑎−𝑁_𝐶𝐴()(𝑎,𝑏=𝐴𝑎,𝑏)1+(2𝜇−1)²𝑣²−12𝐴((𝑟)+𝑟𝑎,𝑏)cosh(𝑟)=1sinh2𝐴()×𝑎,𝑏2+2(2𝜇−1)²((𝑟))(𝑟)−𝑟cosh−1−cosh(𝑟)=1sinh2𝐴(𝑎,𝑏)[cosh(𝑟)−1]2(2𝜇−1)²−1+𝜑₁(𝑟,𝐶𝑎)𝜇𝑎+1−𝜇𝑏,𝜇𝑏+1−𝜇−𝑁_𝐴𝐶()(𝑎,𝑏=𝐴𝑎,𝑏)1+(2𝜇−1)²𝑣²−12𝐴(𝑠𝑎,𝑏)1+(𝑠)(𝑠)=1sincos2𝐴()×𝑎,𝑏1+2(2𝜇−1)²((𝑠))−𝑠sec−1(𝑠)(𝑠)=1sincos2𝐴(𝑎,𝑏)[sec(𝑠)−1]2(2𝜇−1)²−1−𝜑₂(𝑠), (58) where the functions 𝜑₁ and 𝜑₂ are defined as in Lemmas 3 and 4, respectively.
Note that 𝜑₁log2+√3√=1−3log2+√33,𝜑₂𝜋3=4√3𝜋−189. (59)
Therefore, Theorem 11 follows easily from Lemmas 3 and 4 together with (58)-(59).

Theorem 12. Let 𝑆_𝐴𝐻(𝑎,𝑏)=𝑆[𝐴(𝑎,𝑏),𝐻(𝑎,𝑏)], 𝑆_𝐻𝐴(𝑎,𝑏)=𝑆[𝐻(𝑎,𝑏),𝐴(𝑎,𝑏)], 𝑆_𝐴𝐶(𝑎,𝑏)=𝑆[𝐴(𝑎,𝑏),𝐶(𝑎,𝑏)], and 𝑆_𝐶𝐴(𝑎,𝑏)=𝑆[𝐶(𝑎,𝑏),𝐴(𝑎,𝑏)]. Then the inequalities 𝐻(𝑎,𝑏)<𝑆_𝐴𝐻(𝑎,𝑏)<𝑁_𝐴𝐻(𝑎,𝑏)<𝑆_𝐻𝐴(𝑎,𝑏)<𝑁_𝐻𝐴(𝑎,𝑏)<𝐴(𝑎,𝑏)<𝑀(𝑎,𝑏)<𝑇(𝑎,𝑏)<𝑆_𝐶𝐴(𝑎,𝑏)<𝑁_𝐶𝐴()()𝑎,𝑏<𝑄𝑎,𝑏<𝑆_𝐴𝐶(𝑎,𝑏)<𝑁_𝐴𝐶(𝑎,𝑏)<𝐶(𝑎,𝑏) (60) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏.

Proof. It follows from [1, (3.10)] and [11, (4.1)] together with [9, (5), Theorems 2 and 5] that 𝑁(𝑎,𝑏)>𝑎+2𝑏3>SB(𝑎,𝑏),𝐻(𝑎,𝑏)<𝑆_𝐴𝐻(𝑎,𝑏)<𝐿(𝑎,𝑏)<𝑆_𝐻𝐴()()()()()𝑎,𝑏<𝑃𝑎,𝑏<𝐴𝑎,𝑏<𝑀𝑎,𝑏<𝑇𝑎,𝑏<𝑆_𝐶𝐴()()𝑎,𝑏<𝑄𝑎,𝑏<𝑆_𝐴𝐶(𝑎,𝑏)<𝐶(𝑎,𝑏). (61)(62)
Therefore, the second, fourth, ninth, and twelfth inequalities follow from (61) and the first, sixth, seventh, eighth, and eleventh inequalities follow from (62) immediately, while the fifth and thirteenth inequalities can be derived from 𝐻(𝑎,𝑏)<𝐴(𝑎,𝑏)<𝐶(𝑎,𝑏) and the fact that 𝑁_𝐴𝐻(𝑎,𝑏) and 𝑁_𝐴𝐶(𝑎,𝑏) are, respectively, the mean values of 𝐴(𝑎,𝑏), 𝐻(𝑎,𝑏), and 𝐴(𝑎,𝑏), 𝐶(𝑎,𝑏).
Next, we prove the third and tenth inequalities. In fact, the third inequality can be derived from the following inequalities (63) [14, Theorem 1.2] and (64) [9, Theorem 3] together with 𝐴(𝑎,𝑏)>𝐻(𝑎,𝑏). Consider the following: 𝑁_𝐴𝐻()<1𝑎,𝑏2𝐴()+1𝑎,𝑏2𝐻(𝑎,𝑏),𝑆_𝐻𝐴()>2𝑎,𝑏𝜋𝐴()+2𝑎,𝑏1−𝜋𝐻(𝑎,𝑏), (63)(64)
while the tenth inequality can be derived from the inequality 𝑁_𝐶𝐴(𝑎,𝑏)<𝐶^𝛽3(𝑎,𝑏)𝐴^1−𝛽3(𝑎,𝑏) in Theorem 9 and 𝐶(𝑎,𝑏)>𝐴(𝑎,𝑏) together with 𝑄(𝑎,𝑏)=𝐴^1/2(𝑎,𝑏)𝐶^1/2(𝑎,𝑏), where 𝛽₃=[2log(2√3+log(2+√3))−log3]/(2log2)−1=0.4648….

Remark 13. He et al. [14, Theorem 1.2], Xia and Chu [17, Theorem 3.1], and Chu et al. [18, Theorem 2.1] proved that the double inequalities 𝛼₁𝐴(𝑎,𝑏)+1−𝛼₁𝐻(𝑎,𝑏)<𝑁_𝐴𝐻(𝑎,𝑏)<𝛽₁𝐴(𝑎,𝑏)+1−𝛽₁𝐻(),𝛼𝑎,𝑏₂𝐴(𝑎,𝑏)+1−𝛼₂𝐻(𝑎,𝑏)<𝑁_𝐻𝐴(𝑎,𝑏)<𝛽₂𝐴(𝑎,𝑏)+1−𝛽₂𝐻(),𝛼𝑎,𝑏₃𝐴(𝑎,𝑏)+1−𝛼₃𝐻()()𝑎,𝑏<𝐿𝑎,𝑏<𝛽₃𝐴(𝑎,𝑏)+1−𝛽₃𝐻(),𝛼𝑎,𝑏₄𝐴(𝑎,𝑏)+1−𝛼₄𝐻()()𝑎,𝑏<𝑃𝑎,𝑏<𝛽₄𝐴(𝑎,𝑏)+1−𝛽₄𝐻(𝑎,𝑏) (65) hold for all 𝑎,𝑏>0 with 𝑎≠𝑏 if and only if 𝛼₁≤1/3, 𝛽₁≥1/2, 𝛼₂≤2/3, 𝛽₂≥𝜋/4=0.7853…, 𝛼₃≤0, 𝛽₃≥2/3, 𝛼₄≤2/𝜋=0.6366…, and 𝛽₄≥5/6.
From the above results we clearly see that the mean values 𝑁_𝐴𝐻(𝑎,𝑏) and 𝐿(𝑎,𝑏) and 𝑁_𝐻𝐴(𝑎,𝑏) and 𝑃(𝑎,𝑏) are not comparable with each other.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11171307 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.