Open Access

Optimal bounds for two Sándor-type means in terms of power means

Journal of Inequalities and Applications20162016:64

https://doi.org/10.1186/s13660-016-0989-0

Received: 8 July 2015

Accepted: 21 January 2016

Published: 16 February 2016

Abstract

In the article, we prove that the double inequalities \(M_{\alpha }(a,b)< S_{QA}(a,b)< M_{\beta}(a,b)\) and \(M_{\lambda }(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b)\) hold for all \(a, b>0\) with \(a\neq b\) if and only if \(\alpha\leq\log 2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots\) , \(\beta\geq5/3\), \(\lambda\leq4\log2/[4+2\log2-\pi]=1.2351\ldots\) and \(\mu\geq4/3\), where \(S_{QA}(a,b)=A(a,b)e^{Q(a,b)/M(a,b)-1}\) and \(S_{AQ}(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}\) are the Sándor-type means, \(A(a,b)=(a+b)/2\), \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\), \(T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]\), and \(M(a,b)=(a-b)/[2\sinh ^{-1}((a-b)/(a+b))]\) are, respectively, the arithmetic, quadratic, second Seiffert, and Neuman-Sándor means.

Keywords

Schwab-Borchardt meanarithmetic meanquadratic meanNeuman-Sándor meansecond Seiffert meanSándor-type meanpower mean

MSC

26E60

1 Introduction

For \(p\in\mathbb{R}\) and \(a,b>0\) with \(a\neq b\), the pth power mean \(M_{p}(a,b)\) and Schwab-Borchardt mean \(\operatorname{SB}(a,b)\) [1, 2] of a and b are, respectively, given by
$$ M_{p}(a,b)= \textstyle\begin{cases} (\frac{a^{p}+b^{p}}{2} )^{1/p},& p\neq0, \\ \sqrt{ab}, & p=0 \end{cases} $$
(1.1)
and
$$ \operatorname{SB}(a,b)= \textstyle\begin{cases} \frac{\sqrt{b^{2}-a^{2}}}{\cos^{-1}{(a/b)}}, & a< b, \\ \frac{\sqrt{a^{2}-b^{2}}}{\cosh^{-1}{(a/b)}}, & a>b, \end{cases} $$
where \(\cos^{-1}(x)\) and \(\cosh^{-1}(x)=\log(x+\sqrt{x^{2}-1})\) are the inverse cosine and inverse hyperbolic cosine functions, respectively.

It is well known that the power mean \(M_{p}(a,b)\) is continuous and strictly increasing with respect to \(p\in\mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\), the Schwab-Borchardt mean \(\operatorname{SB}(a,b)\) is strictly increasing in both a and b, nonsymmetric and homogeneous of degree 1 with respect to a and b. Many symmetric bivariate means are special cases of the Schwab-Borchardt mean. For example, \(P(a,b)=(a-b)/[2\arcsin((a-b)/(a+b))]=\operatorname{SB}[G(a,b), A(a,b)]\) is the first Seiffert mean, \(T(a,b)=(a-b)/[2\arctan ((a-b)/(a+b))]=\operatorname{SB}[A(a,b), Q(a,b)]\) is the second Seiffert mean, \(M(a,b)=(a-b)/[2\sinh^{-1}((a-b)/(a+b))]=\operatorname{SB}[Q(a,b), A(a,b)]\) is the Neuman-Sándor mean, \(L(a,b)=(a-b)/[2\tanh ^{-1}((a-b)/(a+b))]=\operatorname{SB}[A(a,b), G(a,b)]\) is the logarithmic mean, where \(\sinh^{-1}(x)=\log(x+\sqrt{1+x^{2}})\) is the inverse hyperbolic sine function, \(\tanh^{-1}(x)=\log[(1+x)/(1-x)]/2\) is the inverse hyperbolic tangent function, and \(G(a,b)=\sqrt{ab}\), \(A(a,b)=(a+b)/2\), and \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\) are the geometric, arithmetic, and quadratic means of a and b, respectively.

The Sándor mean \(X(a,b)=A(a,b)e^{G(a,b)/P(a,b)-1}\) [3] can be rewritten as \(X(a,b)= A(a,b)e^{G(a,b)/\operatorname{SB}[G(a,b), A(a,b)]-1}\). Yang [4] proved that \(S(a,b)=be^{a/\operatorname{SB}(a,b)-1}\) is a mean of a and b, and introduced two Sándor-type means \(S_{QA}(a,b)\) and \(S_{AQ}(a,b)\) as follows:
$$\begin{aligned}& S_{QA}(a,b)\triangleq S\bigl[Q(a,b), A(a,b)\bigr] \\& \hphantom{S_{QA}(a,b)}=A(a,b)e^{Q(a,b)/\operatorname{SB}[Q(a,b), A(a,b)]-1}=A(a,b)e^{Q(a,b)/M(a,b)-1}, \end{aligned}$$
(1.2)
$$\begin{aligned}& S_{AQ}(a,b)\triangleq S\bigl[A(a,b), Q(a,b)\bigr] \\& \hphantom{S_{AQ}(a,b)}=Q(a,b)e^{A(a,b)/\operatorname{SB}[A(a,b), Q(a,b)]-1}=Q(a,b)e^{A(a,b)/T(a,b)-1}. \end{aligned}$$
(1.3)

Recently, the bounds involving the power mean for certain bivariate means and Gaussian hypergeometric function have attracted the attention of many researchers [521].

Radó [22] (see also [2325]) proved that the double inequalities
$$ M_{p}(a,b)< L(a,b)< M_{q}(a,b),\qquad M_{\lambda}(a,b)< I(a,b)< M_{\mu}(a,b) $$
hold for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq0\), \(q\geq 1/3\), \(\lambda\leq2/3\), and \(\mu\geq\log2\), where \(I(a,b)=(b^{b}/a^{a})^{1/(b-a)}/e\) is the identric mean of a and b.
In [2629], the authors proved that the double inequality
$$ M_{p}(a,b)< T^{\ast}(a,b)< M_{q}(a,b) $$
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq3/2\) and \(q\geq\log2/(\log\pi-\log2)\), where \(T^{\ast}(a,b)=\frac{2}{\pi}\int _{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\, d\theta\) is the Toader mean of a and b.
Jagers [30], Hästö [31, 32], Costin and Toader [33], and Li et al. [34] proved that \(p_{1}=\log2/\log\pi\), \(q_{1}=2/3\), \(p_{2}=\log2/(\log\pi-\log2)\), and \(q_{2}=5/3\) are the best possible parameters such that the double inequalities
$$ M_{p_{1}}(a,b)< P(a,b)< M_{q_{1}}(a,b),\qquad M_{p_{2}}(a,b)< T(a,b)< M_{q_{2}}(a,b) $$
hold for all \(a, b>0\) with \(a\neq b\).
In [3538], the authors proved that the double inequalities
$$\begin{aligned}& M_{\lambda_{1}}(a,b)< M(a,b)< M_{\mu_{1}}(a,b), \\& M_{\lambda_{2}}(a,b)< U(a,b)< M_{\mu_{2}}(a,b), \\& M_{\lambda_{3}}(a,b)< X(a,b)< M_{\mu_{3}}(a,b) \end{aligned}$$
hold for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda_{1}\leq \log2/\log[2\log(1+\sqrt{2})]\), \(\mu_{1}\geq4/3\), \(\lambda_{2}\leq 2\log2/(2\log\pi-\log2)\), \(\mu_{2}\geq4/3\), \(\lambda_{3}\leq1/3\), and \(\mu_{3}\geq\log2/(1+\log2)\), where \(U(a,b)=(a-b)/ [\sqrt {2}\arctan (\frac{a-b}{\sqrt{2ab}} ) ]\) is the Yang mean of a and b.
The main purpose of this paper is to present the best possible parameters α, β, λ, and μ such that the double inequalities
$$ M_{\alpha}(a,b)< S_{QA}(a,b)< M_{\beta}(a,b), \qquad M_{\lambda }(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b) $$
hold for all \(a, b>0\) with \(a\neq b\).

2 Lemmas

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

Lemma 2.1

Let \(p\in\mathbb{R}\) and
$$ f(x)=(p-1)x^{p+1}-3x^{p}+3x^{p-2}+(1-p)x^{p-3}+3x^{2p-2}+x^{2p-3}-x-3. $$
(2.1)
Then the following statements are true:
  1. (1)

    \(f(x)>0\) for all \(x\in(1, \infty)\) if \(p=5/3\);

     
  2. (2)

    there exists \(\sigma\in(1, \infty)\) such that \(f(x)<0\) for \(x\in (1, \sigma)\) and \(f(x)>0\) for \(x\in(\sigma, \infty)\) if \(p=\log2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots\) .

     

Proof

For part (1), if \(p=5/3\), then (2.1) leads to
$$ f(x)=\frac{ (x^{\frac{2}{3}}-1 ) (x^{\frac{1}{3}}-1 )^{2}}{3x^{\frac{4}{3}}} \bigl(2x^{\frac{8}{3}}+4x^{\frac{7}{3}}+8x^{2}+3x^{\frac {5}{3}}+9x^{\frac{4}{3}}+3x+8x^{\frac{2}{3}}+4x^{\frac{1}{3}}+2 \bigr). $$
(2.2)

Therefore, part (1) follows from (2.2).

For part (2), let \(p=\log2/[1+\log2-\sqrt{2}\log(1+\sqrt {2})]=1.5517\ldots\) , \(f_{1}(x)=f^{\prime}(x)\), \(f_{2}(x)=x^{5-p}f^{\prime}_{1}(x)\) and \(f_{3}(x)=f^{\prime}_{2}(x)\). Then simple computations lead to
$$\begin{aligned}& f(1)=0, \qquad \lim_{x\rightarrow+\infty}f(x)=+\infty, \end{aligned}$$
(2.3)
$$\begin{aligned}& f_{1}(1)=12 \biggl(p-\frac{5}{3} \biggr)< 0, \qquad \lim _{x\rightarrow+\infty }f_{1}(x)=+\infty, \end{aligned}$$
(2.4)
$$\begin{aligned}& f_{2}(1)=24 \biggl(p-\frac{3}{2} \biggr) \biggl(p- \frac{5}{3} \biggr)< 0,\qquad \lim_{x\rightarrow+\infty}f_{2}(x)=+ \infty, \end{aligned}$$
(2.5)
$$\begin{aligned}& f_{3}(x)=6 \bigl(p^{2}-1 \bigr) (2p-3)x^{p}+2p(p-2) (2p-3)x^{p-1} \\& \hphantom{f_{3}(x)={}}{}+4p \bigl(p^{2}-1 \bigr)x^{3}-9p(p-1)x^{2}+3(p-2) (p-3). \end{aligned}$$
(2.6)
Note that
$$\begin{aligned}& 2p(p-2) (2p-3)x^{p-1}>2p(p-2) (2p-3)x^{p}, \qquad -9p(p-1)x^{2}>-9p(p-1)x^{3}, \end{aligned}$$
(2.7)
$$\begin{aligned}& p(p-1) (4p-5)x^{3}>p(p-1) (4p-5) \end{aligned}$$
(2.8)
for \(x>1\), and
$$\begin{aligned}& 16p^{3}-32p^{2}+18>16\times1.55^{3}-32 \times1.552^{2}+18=0.503472>0, \end{aligned}$$
(2.9)
$$\begin{aligned}& 4p^{3}-6p^{2}-10p+18>4\times1.5^{3}-6 \times1.6^{2}-10\times1.6+18=0.14>0. \end{aligned}$$
(2.10)
It follows from (2.6)-(2.10) that
$$\begin{aligned} f_{3}(x) >&6 \bigl(p^{2}-1 \bigr) (2p-3)x^{p}+2p(p-2) (2p-3)x^{p} \\ &{}+4p \bigl(p^{2}-1 \bigr)x^{3}-9p(p-1)x^{3}+3(p-2) (p-3) \\ =& \bigl(16p^{3}-32p^{2}+18 \bigr)x^{p}+p(p-1) (4p-5)x^{3}+3(p-2) (p-3) \\ >& \bigl(16p^{3}-32p^{2}+18 \bigr)x^{p}+p(p-1) (4p-5)+3(p-2) (p-3) \\ =& \bigl(16p^{3}-32p^{2}+18 \bigr)x^{p}+ \bigl(4p^{3}-6p^{2}-10p+18 \bigr)>0 \end{aligned}$$
(2.11)
for \(x>1\).

Inequality (2.11) implies that \(f_{2}(x)\) is strictly increasing on \((1, \infty)\). Then from (2.5) we know that there exists \(\sigma_{1}>1\) such that \(f_{1}(x)\) is strictly decreasing on \((1, \sigma_{1}]\) and strictly increasing on \([\sigma_{1}, \infty)\).

It follows from (2.4) and the piecewise monotonicity of \(f_{1}(x)\) that there exists \(\sigma_{2}>1\) such that \(f(x)\) is strictly decreasing on \((1, \sigma_{2}]\) and strictly increasing on \([\sigma_{2}, \infty)\).

Therefore, part (2) follows from (2.3) and the piecewise monotonicity of \(f(x)\). □

Lemma 2.2

Let \(p\in\mathbb{R}\), and
$$ g(x)=(p-1)x^{p+1}-(p+1)x^{p}+(p+1)x^{p-1}+(1-p)x^{p-2}+x^{2p-1}+x^{2p-2}-x-1. $$
(2.12)
Then the following statements are true:
  1. (1)

    \(g(x)>0\) for all \(x\in(1, \infty)\) if \(p=4/3\);

     
  2. (2)

    there exists \(\tau\in(1, \infty)\) such that \(g(x)<0\) for \(x\in (1, \tau)\) and \(g(x)>0\) for \(x\in(\tau, \infty)\) if \(p=4\log2/[4+2\log2-\pi]=1.2351\ldots\) .

     

Proof

For part (1), if \(p=4/3\), then (2.12) becomes
$$ g(x)=\frac{ (x^{1/3}-1 )^{3}}{3x^{2/3}} \bigl(x^{2}+3x^{5/3}+9x^{4/3}+12x+9x^{2/3}+3x^{1/3}+1 \bigr). $$
(2.13)

Therefore, part (1) follows from (2.13).

For part (2), let \(p=4\log2/[4+2\log2-\pi]=1.2351\ldots\) , \(g_{1}(x)=g^{\prime}(x)\), \(g_{2}(x)=x^{4-p}g^{\prime}_{1}(x)/(p-1)\), and \(g_{3}(x)=g^{\prime}_{2}(x)\). Then simple computations lead to
$$\begin{aligned}& g(1)=0, \qquad \lim_{x\rightarrow+\infty}g(x)=+\infty, \end{aligned}$$
(2.14)
$$\begin{aligned}& g_{1}(1)=6 \biggl(p-\frac{4}{3} \biggr)< 0,\qquad \lim _{x\rightarrow+\infty }g_{1}(x)=+\infty, \end{aligned}$$
(2.15)
$$\begin{aligned}& g_{2}(1)=12 \biggl(p-\frac{4}{3} \biggr)< 0, \qquad \lim _{x\rightarrow+\infty }g_{2}(x)=+\infty, \end{aligned}$$
(2.16)
$$\begin{aligned}& g_{3}(x)=2(p+1) (2p-1)x^{p}+2p(2p-3)x^{p-1} \\& \hphantom{g_{3}(x)={}}{}+3p(p+1)x^{2}-2p(p+1)x+(p+1) (p-2). \end{aligned}$$
(2.17)
Note that
$$\begin{aligned}& 2p(2p-3)x^{p-1}>2p(2p-3)x^{p}, \\& 2p(p+1)x< 2p(p+1)x^{2}, \\& (p+1) (p-2)>(p+1) (p-2)x^{2} \end{aligned}$$
(2.18)
for \(x>1\).
It follows from (2.17) and (2.18) that
$$\begin{aligned} g_{3}(x) >&2(p+1) (2p-1)x^{p}+2p(2p-3)x^{p}+3p(p+1)x^{2} \\ &{}-2p(p+1)x^{2}+(p+1) (p-2)x^{2} \\ =&2 \bigl(4p^{2}-2p-1 \bigr)x^{p}+2\bigl(p^{2}-1 \bigr)x^{2}>0 \end{aligned}$$
(2.19)
for \(x>1\).

Inequality (2.19) implies that \(g_{2}(x)\) is strictly increasing on \((1, \infty)\). Then from (2.16) we know that there exists \(\tau_{1}\in(1, \infty)\) such that \(g_{1}(x)\) is strictly decreasing on \((1, \tau_{1}]\) and strictly increasing on \([\tau_{1}, \infty)\).

It follows from (2.15) and the piecewise monotonicity of \(g_{1}(x)\) that there exists \(\tau_{2}\in(1, \infty)\) such that \(g(x)\) is strictly decreasing on \((1, \tau_{2}]\) and strictly increasing on \([\tau_{2}, \infty)\).

Therefore, part (2) follows from (2.14) and the piecewise monotonicity of \(g(x)\). □

3 Main results

Theorem 3.1

The double inequality
$$ M_{\alpha}(a,b)< S_{QA}(a,b)< M_{\beta}(a,b) $$
holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha\leq\log 2/[1+\log2-\log(1+\sqrt{2})]=1.5517\ldots\) and \(\beta\geq5/3\).

Proof

Since both \(S_{QA}(a,b)\) and \(M_{p}(a,b)\) are symmetric and homogeneous of degree one, we assume that \(a>b\). Let \(x=a/b>1\) and \(p>0\). Then (1.1) and (1.2) lead to
$$\begin{aligned}& \log \bigl[S_{QA}(a,b) \bigr]-\log \bigl[M_{p}(a,b) \bigr] \\& \quad =\log \biggl(\frac{x+1}{2} \biggr)+\frac{\sqrt{2(x^{2}+1)}\sinh^{-1} (\frac{x-1}{x+1} )}{x-1}- \frac{1}{p}\log \biggl(\frac {x^{p}+1}{2} \biggr)-1. \end{aligned}$$
(3.1)
Let
$$ F(x)=\log \biggl(\frac{x+1}{2} \biggr)+\frac{\sqrt{2(x^{2}+1)}\sinh ^{-1} (\frac{x-1}{x+1} )}{x-1}- \frac{1}{p}\log \biggl(\frac {x^{p}+1}{2} \biggr)-1. $$
(3.2)
Then elaborated computations lead to
$$\begin{aligned}& F\bigl(1^{+}\bigr)=0, \end{aligned}$$
(3.3)
$$\begin{aligned}& \lim_{x\rightarrow+\infty}F(x)=\sqrt{2}\log(1+\sqrt{2})-(1+\log 2)+ \frac{1}{p}\log2, \end{aligned}$$
(3.4)
$$\begin{aligned}& F^{\prime}(x)=\frac{2(x+1)}{(x-1)^{2}\sqrt{2(x^{2}+1)}}F_{1}(x), \end{aligned}$$
(3.5)
where
$$\begin{aligned}& F_{1}(x)=\frac{\sqrt{2(x^{2}+1)}(x-1)(x^{p-1}+1)}{2(x+1)(x^{p}+1)}-\sinh ^{-1} \biggl( \frac{x-1}{x+1} \biggr), \\& F_{1}(1)=0, \qquad \lim_{x\rightarrow\infty}F_{1}(x)= \frac{\sqrt{2}}{2}-\log (1+\sqrt{2})=-0.1742\ldots< 0, \end{aligned}$$
(3.6)
$$\begin{aligned}& F_{1}^{\prime}(x)=-\frac{x(x-1)}{(x+1)^{2}(x^{p}+1)^{2}\sqrt{2(x^{2}+1)}}f(x), \end{aligned}$$
(3.7)
where \(f(x)\) is defined by (2.1).

We divide the proof into four cases.

Case 1.1. \(p=\log2/[1+\log2-\log(1+\sqrt{2})]\). Then it follows from Lemma 2.1(2) and (3.7) that there exists \(\sigma\in(1, \infty)\) such that \(F_{1}(x)\) is strictly increasing on \((1, \sigma]\) and strictly decreasing on \([\sigma, \infty)\).

Equations (3.5) and (3.6) together with the piecewise monotonicity of \(F_{1}(x)\) lead to the conclusion that there exists \(\sigma_{0}\in(1, \infty)\) such that \(F(x)\) is strictly increasing on \((1, \sigma_{0}]\) and strictly decreasing on \([\sigma_{0}, \infty)\).

Note that (3.4) becomes
$$ \lim_{x\rightarrow+\infty}F(x)=0. $$
(3.8)
Therefore,
$$ S_{QA}(a,b)>M_{\log2/[1+\log2-\log(1+\sqrt{2})]}(a,b) $$
for all \(a, b>0\) with \(a\neq b\) follows from (3.1)-(3.3) and (3.8) together with the piecewise monotonicity of \(F(x)\).

Case 1.2. \(p=5/3\). Then it follows from Lemma 2.1(1) and (3.7) that \(F_{1}(x)\) is strictly decreasing on \((1, \infty)\).

Therefore,
$$ S_{QA}(a,b)< M_{5/3}(a,b) $$
for all \(a, b>0\) with \(a\neq b\) follows from (3.1)-(3.3), (3.5), (3.6), and the monotonicity of \(F(x)\).
Case 1.3. \(p>\log2/[1+\log2-\log(1+\sqrt{2})]\). Then (3.4) leads to
$$ \lim_{x\rightarrow+\infty}F(x)< 0. $$
(3.9)
Equations (3.1) and (3.2) together with inequality (3.9) imply that there exists large enough \(C_{0}>1\) such that
$$ S_{QA}(a,b)< M_{p}(a,b) $$
for all \(a, b>0\) with \(a/b\in(C_{0}, \infty)\).
Case 1.4. \(1< p<5/3\). Let \(x>0\), \(x\rightarrow0\), then making use of (1.1) and (1.2) together with the Taylor expansion we get
$$\begin{aligned}& \begin{aligned}[b] &S_{QA}(1, 1+x)-M_{p}(1,1+x) \\ &\quad = \biggl(1+\frac{x}{2} \biggr)e^{\sqrt{2(x^{2}+2x+2)}\sinh ^{-1}[x/(2+x)]/x-1}- \biggl[ \frac{1+(1+x)^{p}}{2} \biggr]^{1/p} \\ &\quad =\frac{5-3p}{24}x^{2}+o\bigl(x^{2}\bigr). \end{aligned} \end{aligned}$$
(3.10)
Equation (3.10) implies that there exists small enough \(\delta_{0}>0\) such that
$$ S_{QA}(1, 1+x)>M_{p}(1, 1+x) $$
for \(x\in(0, \delta_{0})\).

Therefore, Theorem 3.1 follows easily from Cases 1.1-1.4 and the monotonicity of the function \(p\rightarrow M_{p}(a,b)\). □

Theorem 3.2

The double inequality
$$ M_{\lambda}(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b) $$
holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\lambda\leq4\log 2/[4+2\log2-\pi]=1.2351\ldots\) and \(\beta\geq4/3\).

Proof

Since both \(S_{AQ}(a,b)\) and \(M_{p}(a,b)\) are symmetric and homogeneous of degree one, we assume that \(a>b\). Let \(x=a/b>1\) and \(p>0\). Then (1.1) and (1.3) lead to
$$\begin{aligned}& \log \bigl[S_{AQ}(a,b) \bigr]-\log \bigl[M_{p}(a,b) \bigr] \\& \quad =\frac{1}{2}\log \biggl(\frac{x^{2}+1}{2} \biggr)+ \frac{x+1}{x-1}\arctan \biggl(\frac{x-1}{x+1} \biggr)-\frac{1}{p}\log \biggl(\frac {x^{p}+1}{2} \biggr)-1. \end{aligned}$$
(3.11)
Let
$$ G(x)=\frac{1}{2}\log \biggl(\frac{x^{2}+1}{2} \biggr)+ \frac {x+1}{x-1}\arctan \biggl(\frac{x-1}{x+1} \biggr)-\frac{1}{p}\log \biggl(\frac {x^{p}+1}{2} \biggr)-1. $$
(3.12)
Then elaborated computations lead to
$$\begin{aligned}& G\bigl(1^{+}\bigr)=0, \end{aligned}$$
(3.13)
$$\begin{aligned}& \lim_{x\rightarrow+\infty}G(x)=\frac{\pi}{4}-\frac{1}{2}\log2-1+ \frac {1}{p}\log2, \end{aligned}$$
(3.14)
$$\begin{aligned}& G^{\prime}(x)=\frac{2}{(x-1)^{2}}G_{1}(x), \end{aligned}$$
(3.15)
where
$$\begin{aligned}& G_{1}(x)=\frac{(x-1)(x^{p-1}+1)}{2(x^{p}+1)}-\arctan \biggl(\frac {x-1}{x+1} \biggr), \\& G_{1}(1)=0, \qquad \lim_{x\rightarrow+\infty}G_{1}(x)= \frac{1}{2}-\frac{\pi}{4}< 0, \end{aligned}$$
(3.16)
$$\begin{aligned}& G^{\prime}_{1}(x)=-\frac{x-1}{2(x^{2}+1)^{2}(x^{p}+1)^{2}}g(x), \end{aligned}$$
(3.17)
where \(g(x)\) is defined by (2.12).

We divide the proof into four cases.

Case 2.1. \(p=4\log2/[4+2\log2-\pi]\). Then it follows from Lemma 2.2(2) and (3.17) that there exists \(\tau\in(1, \infty)\) such that \(G_{1}(x)\) is strictly increasing on \((1, \tau]\) and strictly decreasing on \([\tau, \infty)\).

Equations (3.15) and (3.16) together with the piecewise monotonicity of \(G_{1}(x)\) lead to the conclusion that there exists \(\tau_{0}\in(1, \infty)\) such that \(G(x)\) is strictly increasing on \((1, \tau_{0}]\) and strictly decreasing on \([\tau_{0}, \infty)\).

Note that (3.14) becomes
$$ \lim_{x\rightarrow+\infty}G(x)=0. $$
(3.18)
Therefore,
$$ S_{AQ}(a,b)>M_{4\log2/[4+2\log2-\pi]}(a,b) $$
follows from (3.11)-(3.13) and (3.18) together with the piecewise monotonicity of \(G(x)\).

Case 2.2. \(p=4/3\). Then Lemma 2.2(2) and (3.17) imply that \(G_{1}(x)\) is strictly decreasing on \((1, \infty)\).

Therefore,
$$ S_{AQ}(a,b)< M_{4/3}(a,b) $$
follows easily from (3.11)-(3.13), (3.15), (3.16), and the monotonicity of \(G_{1}(x)\).
Case 2.3. \(p>4\log2/[4+2\log2-\pi]\). Then (3.14) leads to
$$ \lim_{x\rightarrow+\infty}G(x)< 0. $$
(3.19)
Equations (3.11) and (3.12) and inequality (3.19) imply that there exists large enough \(C_{1}>1\) such that
$$ S_{AQ}(a,b)< M_{p}(a,b) $$
for all \(a, b>0\) with \(a/b\in(C_{1}, \infty)\).
Case 2.4. \(0< p<4/3\). Let \(x>0\) and \(x\rightarrow0\). Then making use of (1.1) and (1.3) together with the Taylor expansion we get
$$\begin{aligned}& S_{AQ}(1, 1+x)-M_{p}(1,1+x) \\& \quad =\sqrt{\frac{1+(1+x)^{2}}{2}}e^{(2+x)\arctan[x/(2+x)]/x-1}- \biggl[\frac {1+(1+x)^{p}}{2} \biggr]^{1/p} \\& \quad =\frac{4-3p}{24}x^{2}+o\bigl(x^{2}\bigr). \end{aligned}$$
(3.20)
Equation (3.20) implies that there exists small enough \(\delta_{1}>0\) such that
$$ S_{AQ}(1, 1+x)>M_{p}(1,1+x) $$
for \(x\in(0, \delta_{1})\).

Therefore, Theorem 3.2 follows easily from Cases 2.1-2.4 and the monotonicity of the function \(p\rightarrow M_{p}(a,b)\). □

Declarations

Acknowledgements

The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions. The research was supported by the Natural Science Foundation of China under Grants 11301127, 11371125 and 61374086, and the Natural Science Foundation of Hunan Province under Grant 12C0577.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Hangzhou Normal University, Hangzhou, China
(2)
School of Distance Education, Huzhou Broadcast and TV University, Huzhou, China
(3)
School of Mathematics and Computation Sciences, Hunan City University, Yiyang, China

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