Note on certain inequalities for Neuman means

Abstract

In this paper, we give the explicit formulas for the Neuman means $N_{AH}$, $N_{HA}$, $N_{AC}$, and $N_{CA}$, and present the best possible upper and lower bounds for these means in terms of the combinations of harmonic mean H, arithmetic mean A, and contraharmonic mean C.

MSC:26E60.

1 Introduction

Let $a,b,c\ge 0$ with $ab+ac+bc\ne 0$. Then the symmetric integral $R_F(a,b,c)$ [1] of the first kind is defined as

$$R_F(a,b,c)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}{[(t+a)(t+b)(t+c)]}^{-1/2}dt.$$

The degenerate case of $R_F$, denoted by $R_C$, plays an important role in the theory of special functions [1, 2], which is given by

$$R_C(a,b)=R_F(a,b,b).$$

For $a,b>0$ with $a\ne b$, the Schwab-Borchardt mean $SB(a,b)$ [3–5] of a and b is given by

$$SB(a,b)=\{\begin{array}{cc}\frac{\sqrt{b^2-a^2}}{{cos}^{-1}(a/b)},\hfill & a<b,\hfill \\ \frac{\sqrt{a^2-b^2}}{{cosh}^{-1}(a/b)},\hfill & a>b,\hfill \end{array}$$

where ${cos}^{-1}(x)$ and ${cosh}^{-1}(x)=log(x+\sqrt{x^2-1})$ are the inverse cosine and inverse hyperbolic cosine functions, respectively.

Carlson [6] (see also [[7], (3.21)]) proved that

$$SB(a,b)={\left[R_C(a^2,b^2)\right]}^{-1}.$$

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 [3–5, 8–11].

Let $a>b>0$, $v=(a-b)/(a+b)\in (0,1)$, $p\in (0,\mathrm{\infty })$, $q\in (0,\pi /2)$, $r\in (0,log(2+\sqrt{3}))$, and $s\in (0,\pi /3)$ be the parameters such that $1/cosh(p)=cos(q)=1-v^2$, $cosh(r)=sec(s)=1+v^2$, $H(a,b)=2ab/(a+b)$, $G(a,b)=\sqrt{ab}$, $A(a,b)=(a+b)/2$, $Q(a,b)=\sqrt{(a^2+b^2)/2}$, and $C(a,b)=(a^2+b^2)/(a+b)$ be, respectively, the harmonic, geometric, arithmetic, quadratic, and contraharmonic means of a and b, $S_{AH}(a,b)=SB[A(a,b),H(a,b)]$, $S_{HA}(a,b)=SB[H(a,b),A(a,b)]$, $S_{AC}(a,b)=SB[A(a,b),C(a,b)]$, $S_{CA}(a,b)=SB[C(a,b),A(a,b)]$. Then Neuman [10] gave the explicit formulas

$$S_{AH}(a,b)=A(a,b)\frac{tanh(p)}{p},S_{HA}(a,b)=A(a,b)\frac{sinq}{q},$$

(1.1)

$$S_{CA}(a,b)=A(a,b)\frac{sinh(r)}{r},S_{AC}(a,b)=A(a,b)\frac{tans}{s}.$$

(1.2)

Very recently, Neuman [12] found a new mean $N(a,b)$ derived from the Schwab-Borchardt mean as follows:

$$N(a,b)=\frac{1}{2}[a+\frac{b^2}{SB(a,b)}].$$

(1.3)

Let $N_{AH}(a,b)=N[A(a,b),H(a,b)]$, $N_{HA}(a,b)=N[H(a,b),A(a,b)]$, $N_{AG}(a,b)=N[A(a,b),G(a,b)]$, $N_{GA}(a,b)=N[G(a,b),A(a,b)]$, $N_{AC}(a,b)=N[A(a,b),C(a,b)]$, $N_{CA}(a,b)=N[C(a,b),A(a,b)]$, $N_{AQ}(a,b)=N[A(a,b),Q(a,b)]$, and $N_{QA}(a,b)=N[Q(a,b),A(a,b)]$ be the Neuman means. Then Neuman [12] proved that

$$G(a,b)<N_{AG}(a,b)<N_{GA}(a,b)<A(a,b)<N_{QA}(a,b)<N_{AQ}(a,b)<Q(a,b)$$

for all $a,b>0$ with $a\ne b$, and the double inequalities

$$\begin{array}{r}{\alpha }_{1}A(a,b)+(1-{\alpha }_1)G(a,b)<N_{GA}(a,b)<{\beta }_{1}A(a,b)+(1-{\beta }_1)G(a,b),\\ {\alpha }_{2}Q(a,b)+(1-{\alpha }_2)A(a,b)<N_{AQ}(a,b)<{\beta }_{2}Q(a,b)+(1-{\beta }_2)A(a,b),\\ {\alpha }_{3}A(a,b)+(1-{\alpha }_3)G(a,b)<N_{AG}(a,b)<{\beta }_{3}A(a,b)+(1-{\beta }_3)G(a,b),\\ {\alpha }_{4}Q(a,b)+(1-{\alpha }_4)A(a,b)<N_{QA}(a,b)<{\beta }_{4}Q(a,b)+(1-{\beta }_4)A(a,b)\end{array}$$

hold for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_1\le 2/3$, ${\beta }_1\ge \pi /4$, ${\alpha }_2\le 2/3$, ${\beta }_2\ge (\pi -2)/[4(\sqrt{2}-1)]=0.689\dots$ , ${\alpha }_3\le 1/3$, ${\beta }_3\ge 1/2$, ${\alpha }_4\le 1/3$, and ${\beta }_4\ge [log(1+\sqrt{2})+\sqrt{2}-2]/[2(\sqrt{2}-1)]=0.356\dots$ .

Zhang et al. [13] presented the best possible parameters ${\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2\in [0,1/2]$ and ${\alpha }_3,{\alpha }_4,{\beta }_3,{\beta }_4\in [1/2,1]$ such that the double inequalities

$$\begin{array}{c}G({\alpha }_{1}a+(1-{\alpha }_1)b,{\alpha }_{1}b+(1-{\alpha }_1)a)<N_{AG}(a,b)<G({\beta }_{1}a+(1-{\beta }_1)b,{\beta }_{1}b+(1-{\beta }_1)a),\hfill \\ G({\alpha }_{2}a+(1-{\alpha }_2)b,{\alpha }_{2}b+(1-{\alpha }_2)a)<N_{GA}(a,b)<G({\beta }_{2}a+(1-{\beta }_2)b,{\beta }_{2}b+(1-{\beta }_2)a),\hfill \\ Q({\alpha }_{3}a+(1-{\alpha }_3)b,{\alpha }_{3}b+(1-{\alpha }_3)a)<N_{QA}(a,b)<Q({\beta }_{3}a+(1-{\beta }_3)b,{\beta }_{3}b+(1-{\beta }_3)a),\hfill \\ Q({\alpha }_{4}a+(1-{\alpha }_4)b,{\alpha }_{4}b+(1-{\alpha }_4)a)<N_{AQ}(a,b)<Q({\beta }_{4}a+(1-{\beta }_4)b,{\beta }_{4}b+(1-{\beta }_4)a)\hfill \end{array}$$

hold for all $a,b>0$ with $a\ne b$.

In [14], the authors found the greatest values ${\alpha }_1$, ${\alpha }_2$, ${\alpha }_3$, ${\alpha }_4$, ${\alpha }_5$, ${\alpha }_6$, ${\alpha }_7$, ${\alpha }_8$, and the least values ${\beta }_1$, ${\beta }_2$, ${\beta }_3$, ${\beta }_4$, ${\beta }_5$, ${\beta }_6$, ${\beta }_7$, ${\beta }_8$ such that the double inequalities

$$\begin{array}{c}{A}^{{\alpha }_1}(a,b)G^{1-{\alpha }_1}(a,b)<N_{GA}(a,b)<A^{{\beta }_1}(a,b)G^{1-{\beta }_1}(a,b),\hfill \\ \frac{{\alpha }_2}{G(a,b)}+\frac{1-{\alpha }_2}{A(a,b)}<\frac{1}{N_{GA}(a,b)}<\frac{{\beta }_2}{G(a,b)}+\frac{1-{\beta }_2}{A(a,b)},\hfill \\ A^{{\alpha }_3}(a,b)G^{1-{\alpha }_3}(a,b)<N_{AG}(a,b)<A^{{\beta }_3}(a,b)G^{1-{\beta }_3}(a,b),\hfill \\ \frac{{\alpha }_4}{G(a,b)}+\frac{1-{\alpha }_4}{A(a,b)}<\frac{1}{N_{AG}(a,b)}<\frac{{\beta }_4}{G(a,b)}+\frac{1-{\beta }_4}{A(a,b)},\hfill \\ Q^{{\alpha }_5}(a,b)A^{1-{\alpha }_5}(a,b)<N_{AQ}(a,b)<Q^{{\beta }_5}(a,b)A^{1-{\beta }_5}(a,b),\hfill \\ \frac{{\alpha }_6}{A(a,b)}+\frac{1-{\alpha }_6}{Q(a,b)}<\frac{1}{N_{AQ}(a,b)}<\frac{{\beta }_6}{A(a,b)}+\frac{1-{\beta }_6}{Q(a,b)},\hfill \\ Q^{{\alpha }_7}(a,b)A^{1-{\alpha }_7}(a,b)<N_{QA}(a,b)<Q^{{\beta }_7}(a,b)A^{1-{\beta }_7}(a,b),\hfill \\ \frac{{\alpha }_8}{A(a,b)}+\frac{1-{\alpha }_8}{Q(a,b)}<\frac{1}{N_{QA}(a,b)}<\frac{{\beta }_8}{A(a,b)}+\frac{1-{\beta }_8}{Q(a,b)}\hfill \end{array}$$

hold for all $a,b>0$ with $a\ne b$.

The main purpose of this paper is to give the explicit formulas for the Neuman means $N_{AH}$, $N_{HA}$, $N_{AC}$, and $N_{CA}$, and to present the best possible upper and lower bounds for these means in terms of the combinations of harmonic, arithmetic, and contraharmonic means. Our main results are Theorems 1.1-1.3.

Theorem 1.1 Let $a>b>0$, $v=(a-b)/(a+b)\in (0,1)$, $p\in (0,\mathrm{\infty })$, $q\in (0,\pi /2)$, $r\in (0,log(2+\sqrt{3}))$, and $s\in (0,\pi /3)$ be the parameters such that $1/cosh(p)=cos(q)=1-v^2$, $cosh(r)=sec(s)=1+v^2$. Then we have

$$N_{AH}(a,b)=\frac{1}{2}A(a,b)[1+\frac{2p}{sinh(2p)}],$$

(1.4)

$$N_{HA}(a,b)=\frac{1}{2}A(a,b)[cos(q)+\frac{q}{sin(q)}],$$

(1.5)

$$N_{CA}(a,b)=\frac{1}{2}A(a,b)[cosh(r)+\frac{r}{sinh(r)}],$$

(1.6)

$$N_{AC}(a,b)=\frac{1}{2}A(a,b)[1+\frac{2s}{sin(2s)}],$$

(1.7)

and

$$\begin{array}{rl}H(a,b)& <N_{AH}(a,b)<N_{HA}(a,b)<A(a,b)\\ <N_{CA}(a,b)<N_{AC}(a,b)<C(a,b).\end{array}$$

(1.8)

Theorem 1.2 The double inequalities

$${\alpha }_{1}A(a,b)+(1-{\alpha }_1)H(a,b)<N_{AH}(a,b)<{\beta }_{1}A(a,b)+(1-{\beta }_1)H(a,b),$$

(1.9)

$${\alpha }_{2}A(a,b)+(1-{\alpha }_2)H(a,b)<N_{HA}(a,b)<{\beta }_{2}A(a,b)+(1-{\beta }_2)H(a,b),$$

(1.10)

$${\alpha }_{3}C(a,b)+(1-{\alpha }_3)A(a,b)<N_{CA}(a,b)<{\beta }_{3}C(a,b)+(1-{\beta }_3)A(a,b),$$

(1.11)

$${\alpha }_{4}C(a,b)+(1-{\alpha }_4)A(a,b)<N_{AC}(a,b)<{\beta }_{4}C(a,b)+(1-{\beta }_4)A(a,b)$$

(1.12)

hold for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_1\le 1/3$, ${\beta }_1\ge 1/2$, ${\alpha }_2\le 2/3$, ${\beta }_2\ge \pi /4=0.7853\dots$ , ${\alpha }_3\le 1/3$, ${\beta }_3\ge \sqrt{3}log(2+\sqrt{3})/6=0.3801\dots$ , ${\alpha }_4\le 2/3$, and ${\beta }_4\ge (4\sqrt{3}\pi -9)/18=0.7901\dots$ .

Theorem 1.3 The double inequalities

$$\frac{{\alpha }_5}{H(a,b)}+\frac{1-{\alpha }_5}{A(a,b)}<\frac{1}{N_{AH}(a,b)}<\frac{{\beta }_5}{H(a,b)}+\frac{1-{\beta }_5}{A(a,b)},$$

(1.13)

$$\frac{{\alpha }_6}{H(a,b)}+\frac{1-{\alpha }_6}{A(a,b)}<\frac{1}{N_{HA}(a,b)}<\frac{{\beta }_6}{H(a,b)}+\frac{1-{\beta }_6}{A(a,b)},$$

(1.14)

$$\frac{{\alpha }_7}{A(a,b)}+\frac{1-{\alpha }_7}{C(a,b)}<\frac{1}{N_{CA}(a,b)}<\frac{{\beta }_7}{A(a,b)}+\frac{1-{\beta }_7}{C(a,b)},$$

(1.15)

$$\frac{{\alpha }_8}{A(a,b)}+\frac{1-{\alpha }_8}{C(a,b)}<\frac{1}{N_{AC}(a,b)}<\frac{{\beta }_8}{A(a,b)}+\frac{1-{\beta }_8}{C(a,b)},$$

(1.16)

hold for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_5\le 0$, ${\beta }_5\ge 2/3$, ${\alpha }_6\le 0$, ${\beta }_6\ge 1/3$, ${\alpha }_7\le [2\sqrt{3}-log(2+\sqrt{3})]/[2\sqrt{3}+log(2+\sqrt{3})]=0.4490\dots$ , ${\beta }_7\ge 2/3$, ${\alpha }_8\le (9\sqrt{3}-4\pi )/(3\sqrt{3}+4\pi )=0.1701\dots$ , and ${\beta }_8\ge 1/3$.

2 Lemmas

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

Lemma 2.1 (See [[15], Theorem 1.25])

For $-\mathrm{\infty }<a<b<\mathrm{\infty }$, let $f,g:[a,b]\to \mathbb{R}$ be continuous on $[a,b]$, and be differentiable on $(a,b)$, let $g^{\prime }(x)\ne 0$ on $(a,b)$. If $f^{\prime }(x)/g^{\prime }(x)$ is increasing (decreasing) on $(a,b)$, then so are

$$\frac{f(x)-f(a)}{g(x)-g(a)}\mathit{\text{and}}\frac{f(x)-f(b)}{g(x)-g(b)}.$$

If $f^{\prime }(x)/g^{\prime }(x)$ is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2.2 (See [[16], Lemma 1.1])

Suppose that the power series $f(x)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{x}^n$ and $g(x)={\sum }_{n=0}^{\mathrm{\infty }}{b}_n{x}^n$ have the radius of convergence $r>0$ and $a_n,b_n>0$ for all $n\ge 0$. If the sequence $\{a_n/b_n\}$ is (strictly) increasing (decreasing) for all $n\ge 0$, then the function $f(x)/g(x)$ is also (strictly) increasing (decreasing) on $(0,r)$.

Lemma 2.3 (See [[12], Theorem 4.1])

If $a>b$, then

$$N(b,a)>N(a,b).$$

Lemma 2.4 The function

$${\phi }_1(t)=\frac{sinh(2t)-4sinh(t)+2t}{sinh(2t)-2sinh(t)}$$

is strictly increasing from $(0,\mathrm{\infty })$ onto $(2/3,1)$.

Proof Making use of power series expansion we get

$${\phi }_1(t)=\frac{{\sum }_{n=1}^{\mathrm{\infty }}\frac{2^{2n+1}-4}{(2n+1)!}{t}^{2n+1}}{{\sum }_{n=1}^{\mathrm{\infty }}\frac{2^{2n+1}-2}{(2n+1)!}{t}^{2n+1}}=\frac{{\sum }_{n=0}^{\mathrm{\infty }}\frac{2^{2n+3}-4}{(2n+3)!}{t}^{2n}}{{\sum }_{n=0}^{\mathrm{\infty }}\frac{2^{2n+3}-2}{(2n+3)!}{t}^{2n}}.$$

(2.1)

Let

$$a_n=\frac{2^{2n+3}-4}{(2n+3)!},b_n=\frac{2^{2n+3}-2}{(2n+3)!}.$$

(2.2)

Then

$$a_n>0,b_n>0,$$

(2.3)

and $a_n/b_n=1-1/(2^{2n+2}-1)$ is strictly increasing for all $n\ge 0$.

Note that

$${\phi }_1\left(0^{+}\right)=\frac{a_0}{b_0}=\frac{2}{3},{\phi }_1(\mathrm{\infty })=\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}=1.$$

(2.4)

Therefore, Lemma 2.4 follows easily from Lemma 2.2 and (2.1)-(2.4) together with the monotonicity of the sequence $\{a_n/b_n\}$. □

Lemma 2.5 The function

$${\phi }_2(t)=\frac{2t-sin(2t)}{sint(1-cost)}$$

is strictly increasing from $(0,\pi /2)$ onto $(8/3,\pi )$.

Proof Let $f_1(t)=2t-sin(2t)$ and $g_1(t)=sint(1-cost)$. Then simple computations lead to

$${\phi }_2(t)=\frac{f_1(t)-f_1(0)}{g_1(t)-g_1(0)}$$

(2.5)

and $f_1^{\prime }(t)/g_1^{\prime }(t)=4[1-1/(2+1/cost)]$ is strictly increasing on $(0,\pi /2)$.

Note that

$${\phi }_2\left(0^{+}\right)=\underset{t\to 0^{+}}{lim}\frac{f_1^{\prime }(t)}{g_1^{\prime }(t)}=\frac{8}{3},{\phi }_2(\pi /2)=\pi .$$

(2.6)

Therefore, Lemma 2.5 follows from Lemma 2.1, (2.5), (2.6), and the monotonicity of $f_1^{\prime }(t)/g_1^{\prime }(t)$. □

Lemma 2.6 The function

$${\phi }_3(t)=\frac{sinh(t)cosh(t)-t}{[sinh(t)cosh(t)+t](cosh(t)-1)}$$

is strictly decreasing from $(0,\mathrm{\infty })$ onto $(0,2/3)$.

Proof Simple computations lead to

$$\begin{array}{rl}{\phi }_3(t)& =\frac{2sinh(2t)-4t}{sinh(3t)+4tcosh(t)+sinh(t)-2sinh(2t)-4t}\\ =\frac{{\sum }_{n=0}^{\mathrm{\infty }}\frac{2^{2n+4}}{(2n+3)!}{t}^{2n}}{{\sum }_{n=0}^{\mathrm{\infty }}\frac{3^{2n+3}-2^{2n+4}+8n+13}{(2n+3)!}{t}^{2n}}.\end{array}$$

(2.7)

Let

$$a_n=\frac{2^{2n+4}}{(2n+3)!},b_n=\frac{3^{2n+3}-2^{2n+4}+8n+13}{(2n+3)!}.$$

(2.8)

Then

$$a_n>0,b_n>0,$$

(2.9)

and

$$\frac{a_{n+1}}{b_{n+1}}-\frac{a_n}{b_n}=-\frac{2^{2n+4}(5\times 3^{2n+3}-24n-31)}{(3^{2n+5}-2^{2n+6}+8n+21)(3^{2n+3}-2^{2n+4}+8n+13)}<0$$

(2.10)

for all $n\ge 0$.

Note that

$${\phi }_3\left(0^{+}\right)=\frac{a_0}{b_0}=\frac{2}{3},{\phi }_3(\mathrm{\infty })=\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}=0.$$

(2.11)

Therefore, Lemma 2.6 follows easily from (2.7)-(2.11) and Lemma 2.2. □

Lemma 2.7 The function

$$f(t)=9cost+\frac{t}{sint}$$

is strictly decreasing on the interval $(0,\pi /2)$.

Proof Let $f_2(t)=9sintcost+t$ and $g_2(t)=sint$. Then simple computations lead to

$$\begin{array}{r}f(t)=\frac{f_2(t)-f_2(0)}{g_2(t)-g_2(0)},\\ \frac{f_2^{\prime }(t)}{g_2^{\prime }(t)}=\frac{18{cos}^{2}t-8}{cost},\end{array}$$

(2.12)

and

$${\left[\frac{f_2^{\prime }(t)}{g_2^{\prime }(t)}\right]}^{\prime }=-\frac{2sint(9{cos}^{2}t+4)}{{cos}^{2}t}<0$$

(2.13)

for $t\in (0,\pi /2)$.

Therefore, Lemma 2.7 follows easily from (2.12) and (2.13) together with Lemma 2.1. □

Lemma 2.8 The function

$${\phi }_4(t)=\frac{sintcost-t}{(t+sintcost)(1-cost)}$$

is strictly decreasing from $(0,\pi /2)$ onto $(-1,-2/3)$.

Proof Let $f_3(t)=sintcost-t$ and $g_3(t)=(t+sintcost)(1-cost)$. Then simple computations lead to

$${\phi }_4(t)=\frac{f_3(t)}{g_3(t)}=\frac{f_3(t)-f_3(0)}{g_3(t)-g_3(0)},$$

(2.14)

$$\frac{f_3^{\prime }(t)}{g_3^{\prime }(t)}=\frac{f_3^{\prime }(t)-f_3^{\prime }(0)}{g_3^{\prime }(t)-g_3^{\prime }(0)},$$

(2.15)

and

$$\frac{f_3^{″}(t)}{g_3^{″}(t)}=\frac{4}{4-(9cost+\frac{t}{sint})}.$$

(2.16)

Note that

$${\phi }_4\left(0^{+}\right)=\underset{t\to 0^{+}}{lim}\frac{f_3^{″}(t)}{g_3^{″}(t)}=-\frac{2}{3},{\phi }_4\left(\frac{\pi }{2}\right)=-1.$$

(2.17)

Therefore, Lemma 2.8 follows from Lemma 2.1 and Lemma 2.7 together with (2.14)-(2.17). □

3 Proofs of Theorems 1.1-1.3

Proof of Theorem 1.1 It follows from (1.1)-(1.3) as we clearly see that

$$\begin{array}{c}\begin{array}{rl}{N}_{AH}(a,b)& =\frac{1}{2}[A(a,b)+\frac{H^2(a,b)}{S_{AH}(a,b)}]=\frac{1}{2}A(a,b)[1+{(1-v^2)}^2\frac{p}{tanh(p)}]\\ =\frac{1}{2}A(a,b)[1+\frac{p}{tanh(p){cosh}^2(p)}]=\frac{1}{2}A(a,b)[1+\frac{2p}{sinh(2p)}],\end{array}\hfill \\ \begin{array}{rl}{N}_{HA}(a,b)& =\frac{1}{2}[H(a,b)+\frac{A^2(a,b)}{S_{HA}(a,b)}]=\frac{1}{2}A(a,b)[(1-v^2)+\frac{q}{sinq}]\\ =\frac{1}{2}A(a,b)[cosq+\frac{q}{sinq}],\end{array}\hfill \\ \begin{array}{rl}{N}_{CA}(a,b)& =\frac{1}{2}[C(a,b)+\frac{A^2(a,b)}{S_{CA}(a,b)}]=\frac{1}{2}A(a,b)[(1+v^2)+\frac{r}{sinh(r)}]\\ =\frac{1}{2}A(a,b)[cosh(r)+\frac{r}{sinh(r)}],\end{array}\hfill \\ \begin{array}{rl}{N}_{AC}(a,b)& =\frac{1}{2}[A(a,b)+\frac{C^2(a,b)}{S_{AC}(a,b)}]=\frac{1}{2}A(a,b)[1+{(1+v^2)}^2\frac{s}{tan(s)}]\\ =\frac{1}{2}A(a,b)[1+\frac{s}{tan(s){cos}^{2}s}]=\frac{1}{2}A(a,b)[1+\frac{2s}{sin(2s)}].\end{array}\hfill \end{array}$$

Inequalities (1.8) follow easily from $H(a,b)<A(a,b)<C(a,b)$ and Lemma 2.3 together with the fact that $N_{KL}(a,b)$ is a mean of $K(a,b)$ and $L(a,b)$ for $K(a,b),L(a,b)\in \{H(a,b),A(a,b),C(a,b)\}$. □

Proof of Theorem 1.2 Without loss of generality, we assume that $a>b$. Let $v=(a-b)/(a+b)\in (0,1)$, $p\in (0,\mathrm{\infty })$, $q\in (0,\pi /2)$, $r\in (0,log(2+\sqrt{3}))$, and $s\in (0,\pi /3)$ be the parameters such that $1/cosh(p)=cos(q)=1-v^2$, $cosh(r)=sec(s)=1+v^2$. Then from (1.4)-(1.7) we have

$$\begin{array}{rl}\frac{N_{AH}(a,b)-H(a,b)}{A(a,b)-H(a,b)}& =\frac{[1+2p/sinh(2p)]/2-(1-v^2)}{v^2}\\ =\frac{[1+2p/sinh(2p)]/2-1/cosh(p)}{1-1/cosh(p)}={\phi }_1(p),\end{array}$$

(3.1)

$$\begin{array}{rl}\frac{N_{HA}(a,b)-H(a,b)}{A(a,b)-H(a,b)}& =\frac{[cosq+q/sinq]/2-(1-v^2)}{v^2}\\ =\frac{[cosq+q/sinq]/2-cosq}{1-cosq}=\frac{1}{4}{\phi }_2(q),\end{array}$$

(3.2)

$$\begin{array}{rl}\frac{N_{CA}(a,b)-A(a,b)}{C(a,b)-A(a,b)}& =\frac{[cosh(r)+r/sinh(r)]/2-1}{v^2}\\ =\frac{[cosh(r)+r/sinh(r)]/2-1}{cosh(r)-1}=\frac{1}{2}{\phi }_1(r),\end{array}$$

(3.3)

$$\begin{array}{rl}\frac{N_{AC}(a,b)-A(a,b)}{C(a,b)-A(a,b)}& =\frac{[1+2s/sin(2s)]/2-1}{v^2}\\ =\frac{[1+2s/sin(2s)]/2-1}{sec(s)-1}=\frac{1}{4}{\phi }_2(s),\end{array}$$

(3.4)

where the functions ${\phi }_1$ and ${\phi }_2$ are defined as in Lemmas 2.4 and 2.5, respectively.

Note that

$${\phi }_1[log(2+\sqrt{3})]=\sqrt{3}log(2+\sqrt{3})/6$$

(3.5)

and

$${\phi }_2\left(\frac{\pi }{3}\right)=\frac{8\sqrt{3}\pi -18}{9}.$$

(3.6)

Therefore, inequality (1.9) holds for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_1\le 1/3$ and ${\beta }_1\ge 1/2$ follows from (3.1) and Lemma 2.4, inequality (1.10) holds for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_2\le 2/3$ and ${\beta }_2\ge \pi /4$ follows from (3.2) and Lemma 2.5, inequality (1.11) holds for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_3\le 1/3$ and ${\beta }_3\ge \sqrt{3}log(2+\sqrt{3})/6$ follows from (3.3) and (3.5) together with Lemma 2.4, and inequality (1.12) holds for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_4\le 2/3$ and ${\beta }_4\ge (4\sqrt{3}\pi -9)/18$ follows from (3.4) and (3.6) together with Lemma 2.5. □

Proof of Theorem 1.3 Without loss of generality, we assume that $a>b$. Let $v=(a-b)/(a+b)\in (0,1)$, $p\in (0,\mathrm{\infty })$, $q\in (0,\pi /2)$, $r\in (0,log(2+\sqrt{3}))$, and $s\in (0,\pi /3)$ be the parameters such that $1/cosh(p)=cos(q)=1-v^2$, $cosh(r)=sec(s)=1+v^2$. Then from (1.4)-(1.7) we have

$$\frac{1/N_{AH}(a,b)-1/A(a,b)}{1/H(a,b)-1/A(a,b)}=\frac{\frac{2}{1+2p/sinh(2p)}-1}{\frac{1}{1-v^2}-1}=\frac{\frac{2sinh(2p)}{2p+sinh(2p)}-1}{cosh(p)-1}={\phi }_3(p),$$

(3.7)

$$\frac{1/N_{HA}(a,b)-1/A(a,b)}{1/H(a,b)-1/A(a,b)}=\frac{\frac{2}{cos(q)+q/sin(q)}-1}{\frac{1}{1-v^2}-1}=\frac{\frac{2sin(q)-sin(q)cos(q)-q}{sin(q)cos(q)+q}}{\frac{1-cos(q)}{cos(q)}}=1+{\phi }_4(q),$$

(3.8)

$$\frac{1/N_{CA}(a,b)-1/C(a,b)}{1/A(a,b)-1/C(a,b)}=\frac{\frac{2}{cosh(r)+r/sinh(r)}-\frac{1}{1+v^2}}{1-\frac{1}{1+v^2}}={\phi }_3(r),$$

(3.9)

and

$$\frac{1/N_{AC}(a,b)-1/C(a,b)}{1/A(a,b)-1/C(a,b)}=1+{\phi }_4(s),$$

(3.10)

where the functions ${\phi }_3$ and ${\phi }_4$ are defined as in Lemmas 2.6 and 2.8, respectively.

Note that

$${\phi }_3[log(2+\sqrt{3})]=\frac{2\sqrt{3}-log(2+\sqrt{3})}{2\sqrt{3}+log(2+\sqrt{3})}$$

(3.11)

and

$${\phi }_4\left(\frac{\pi }{3}\right)=-\frac{8\pi -6\sqrt{3}}{4\pi +3\sqrt{3}}.$$

(3.12)

Therefore, Theorem 1.3 follows easily from (3.7)-(3.12) together with Lemmas 2.6 and 2.8. □