Optimal two-parameter geometric and arithmetic mean bounds for the Sándor–Yang mean

Abstract

In the article, we provide the sharp bounds for the Sándor–Yang mean in terms of certain families of the two-parameter geometric and arithmetic mean and the one-parameter geometric and harmonic means. As applications, we present new bounds for a certain Yang mean and the inverse tangent function.

1 Introduction

Let $\nu\in(-\infty, \infty)$ and $\sigma, \tau>0$ with $\sigma\neq\tau$. Then we denote by

$$ \begin{gathered} \mathbf{G}(\sigma, \tau)=\sigma^{1/2}\tau^{1/2}, \qquad \mathbf{U}(\sigma, \tau)=\frac{\sqrt{2}(\sigma-\tau)}{2\arctan(\frac {\sqrt{2}(\sigma-\tau)}{2\sqrt{\sigma\tau}} )}, \\\mathbf{Q}(\sigma , \tau)= \biggl(\frac{\sigma^{2}+\tau^{2}}{2} \biggr)^{1/2},\end{gathered} $$

(1.1)

and

$$ \mathbf{H}_{\nu}(\sigma, \tau)= \biggl(\frac{\sigma^{\nu}+\tau^{\nu }}{2} \biggr)^{1/\nu}\quad(\nu\neq0), \qquad\mathbf{ H}_{0}(\sigma, \tau)=\sigma^{1/2}\tau^{1/2} $$

the geometric mean, Yang mean [1], quadratic mean [2], and νth Hölder mean [3] of σ and τ, respectively.

It is not difficult to verify that the νth Hölder mean $H_{\nu }(\sigma, \tau)$ is strictly increasing with respect to $\nu\in(-\infty , \infty)$ for all distinct positive real numbers σ and τ, and

$$\begin{gathered} \mathbf{H}_{-1}(\sigma, \tau)=\frac{2\sigma\tau}{\sigma+\tau}=\mathbf {H}(\sigma, \tau), \qquad\mathbf{H}_{0}(\sigma, \tau)=\sigma^{1/2} \tau^{1/2}=\mathbf{G}(\sigma, \tau), \\ \mathbf{H}_{1}(\sigma, \tau)=\frac{\sigma+\tau}{2}=\mathbf{A}(\sigma, \tau), \qquad\mathbf{H}_{2}(\sigma, \tau)= \biggl(\frac{\sigma^{2}+\tau ^{2}}{2} \biggr)^{1/2}=\mathbf{Q}(\sigma, \tau)\end{gathered} $$

are the classical harmonic, geometric, arithmetic, and quadratic means of σ and τ, respectively.

The bivariate means have in the past decades been the subject of intense research activity [4,5,6,7,8,9,10,11,12,13] because many important special functions can be expressed by the bivariate means [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] and they have wide applications in mathematics, statistics, physics, economics [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55], and many other natural and human social sciences [56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76].

Yang, Wu, and Chu [77] proved that $\kappa_{1}= 2\log2/(2\log\pi-\log 2)\simeq0.8684$ is the largest possible value and $\kappa_{2}=4/3$ is the least possible value such that the two-sided inequality

$$ \mathbf{H}_{\kappa_{1}}(\sigma, \tau)< \mathbf{U}(\sigma, \tau)< \mathbf{H}_{\kappa_{2}}(\sigma, \tau) $$

takes place for all distinct positive real numbers σ and τ, which leads to the conclusion that

$$ \mathbf{G}(\sigma, \tau)< \mathbf{U}(\sigma, \tau)< \mathbf{Q}(\sigma, \tau) $$

for $\sigma, \tau>0$ with $\sigma\neq\tau$.

In [78], Qian and Chu found that $\lambda=\lambda_{0}\simeq0.5451$ and $\mu=2$ are the best possible parameters such that the double inequality

$$ \mathcal{L}_{\lambda}(\sigma, \tau)< \mathbf{U}(\sigma, \tau)< \mathcal{L}_{\mu}(\sigma, \tau) $$

holds for all unequal positive real numbers σ and τ, where

$$\begin{gathered} \mathcal{L}_{\nu}(\sigma, \tau)= \biggl[\frac{\sigma^{\nu+1}-\tau^{\nu +1}}{(\nu+1)(\sigma-\tau)} \biggr]^{1/\nu} \quad(\nu\neq-1, 0) \\ \mathcal{L}_{-1}(\sigma, \tau)=\frac{\sigma-\tau}{\log\sigma-\log\tau}, \qquad \mathcal{L}_{0}(\sigma, \tau)=\frac{1}{e} \biggl( \frac{\sigma^{\sigma}}{\tau^{\tau}} \biggr)^{1/(\sigma-\tau)}\end{gathered} $$

is the νth generalized logarithmic mean of σ and τ.

The Sándor–Yang mean $\mathbf{SY}(\sigma, \tau)$ [1] and two-parameter geometric and arithmetic mean $\mathbf{GA}_{\eta, \nu}(\sigma, \tau)$ [79] are defined by

$$ \mathbf{SY}(\sigma, \tau)=\mathbf{Q}(\sigma, \tau)e^{\mathbf{G}(\sigma, \tau)/\mathbf{U}(\sigma, \tau)-1} $$

(1.2)

and

$$ \mathbf{GA}_{\eta, \nu}(\sigma, \tau)=\mathbf{G}^{\nu} \bigl[\eta \sigma+(1-\eta)\tau, \eta\tau+(1-\eta)\sigma \bigr]\mathbf{A}^{1-\nu}( \sigma, \tau), $$

(1.3)

respectively.

Identity (1.3) leads to the conclusion that

$$\begin{aligned}& \mathbf{GA}_{p, 1}(\sigma, \tau)=\mathbf{G} \bigl[p\sigma+(1-p)\tau, p\tau+(1-p) \sigma \bigr], \end{aligned}$$

(1.4)

$$\begin{aligned}& \mathbf{GA}_{p, 2}(\sigma, \tau)=\mathbf{H} \bigl[p\sigma+(1-p)\tau, p \tau+(1-p)\sigma \bigr], \end{aligned}$$

(1.5)

and

$$ \mathbf{GA}_{p, 0}(\sigma, \tau)=\mathbf{GA}_{1/2, 1/2}(\sigma, \tau)=\mathbf{A}(\sigma, \tau). $$

(1.6)

Chu et al. [79] proved that the inequalities

$$ \mathbf{GA}_{\eta_{1}, \nu}(\sigma, \tau)>\mathbf{AGM}(\sigma, \tau) $$

and

$$ \mathbf{GA}_{\eta_{2}, \nu}(\sigma, \tau)>\mathbf{L}(\sigma, \tau) $$

are valid for all distinct positive real numbers σ and τ if and only if

$$ \eta_{1}\geq\frac{1}{2}-\frac{\sqrt{2\nu}}{4\nu}, \qquad \eta_{2}\geq\frac{1}{2}-\frac{\sqrt{6\nu}}{6\nu} $$

if $\nu\in[1, \infty)$ and $0<\eta_{1}, \eta_{2}<1/2$, where

$$ \mathbf{L}(\sigma, \tau)=\mathcal{L}_{-1}(\sigma, \tau)= \frac{\sigma-\tau}{\log\sigma-\log\tau} $$

and

$$ \mathbf{AGM}(\sigma, \tau)=\frac{\pi}{2\int_{0}^{\pi}\frac{dt}{\sqrt {\sigma^{2}\cos^{2}t+\tau^{2}\sin^{2}t}}} $$

are the logarithmic and Gaussian arithmetic-geometric means of σ and τ, respectively.

Zhang, Yang, and Qian [80], and He et al. [81] proved that

$$ \lambda_{1}=\lambda_{2}=\frac{\sqrt{2}}{e}\simeq0.5203, \qquad\lambda_{3}=\frac{2\log2}{2+\log2}\simeq0.5147, \qquad \nu_{1}=\frac{5}{6}, \qquad\nu_{2}=\nu_{3}= \frac{2}{3} $$

are the best possible parameters such that the double inequalities

$$\begin{gathered} \lambda_{1}\mathbf{A}(\sigma, \tau)+(1-\lambda_{1}) \mathbf{H}(\sigma, \tau)< \mathbf{SY}(\sigma, \tau)< \nu_{1}\mathbf{A}( \sigma, \tau)+(1-\nu_{1})\mathbf{H}(\sigma, \tau), \\ \lambda_{2}\mathbf{A}(\sigma, \tau)+(1-\lambda_{1}) \mathbf{G}(\sigma, \tau)< \mathbf{SY}(\sigma, \tau)< \nu_{2}\mathbf{A}( \sigma, \tau)+(1-\nu_{2})\mathbf{G}(\sigma, \tau),\end{gathered} $$

and

$$ \mathbf{H}_{\lambda_{3}}(\sigma, \tau)< \mathbf{SY}(\sigma, \tau)< \mathbf{H}_{\nu_{3}}(\sigma, \tau) $$

(1.7)

hold for all $\sigma, \tau>0$ with $\sigma\neq\tau$.

From (1.4)–(1.7) and the monotonicity of the function $\nu\rightarrow \mathbf{H}_{\nu}(\sigma, \tau)$, we clearly see that

$$ \begin{aligned}[b] \mathbf{GA}_{1, 2}(\sigma, \tau)&=\mathbf{H}(\sigma, \tau)= \mathbf{H}_{-1}(\sigma, \tau)< \mathbf{G}(\sigma, \tau)= \mathbf{H}_{0}(\sigma, \tau) \\ &< \mathbf{SY}(\sigma, \tau)< \mathbf{H}_{1}(\sigma, \tau)=\mathbf{A}( \sigma, \tau)=\mathbf{GA}_{p, 0}(\sigma, \tau)=\mathbf{GA}_{1/2, 1/2}( \sigma, \tau)\end{aligned} $$

(1.8)

for all $\sigma, \tau>0$ with $\sigma\neq\tau$.

Motivated by inequality (1.8), we naturally ask the question: For fixed $p\in\mathbb{R}$, what are the best possible parameters λ and μ on the interval $(0, 1/2)$ or $(1/2, 1)$ depending only on the parameter p such that the double inequality

$$ \mathbf{GA}_{\lambda, p}(\sigma, \tau)< \mathbf{SY}(\sigma, \tau)< \mathbf{GA}_{\mu, p}(\sigma, \tau) $$

is valid for all unequal positive real numbers σ and τ?

It is the aim of the article to answer the question in the case of $p\in[1, \infty)$ and $\lambda, \mu\in(0, 1/2)$.

2 Lemmas

Lemma 2.1

(see [82, Theorem 1.25])

Let $\kappa_{1}, \kappa_{2}\in\mathbb{R}$with $\kappa_{1}<\kappa _{2}$, $\mathcal{F}, \mathcal{G}: [\kappa_{1}, \kappa_{2}]\rightarrow \mathbb{R}$be continuous on $[\kappa_{1}, \kappa_{2}]$and differentiable on $(\kappa_{1}, \kappa_{2})$with $\mathcal{G}^{\prime }(t)\neq0$on $(\kappa_{1}, \kappa_{2})$. Then both the functions

$$ \frac{\mathcal{F}(t)-\mathcal{F}(\kappa_{1})}{\mathcal{G}(t)-\mathcal {G}(\kappa_{1})} $$

and

$$ \frac{\mathcal{F}(t)-\mathcal{F}(\kappa_{2})}{\mathcal{G}(t)-\mathcal {G}(\kappa_{2})} $$

are (strictly) increasing (decreasing) on $(\kappa_{1}, \kappa_{2})$if $\mathcal{F}^{\prime}(t)/\mathcal{G}^{\prime}(t)$is (strictly) increasing (decreasing) on $(\kappa_{1}, \kappa_{2})$.

Lemma 2.2

The inequality

$$ \frac{1}{3p}+ \biggl(\frac{2}{e^{2}} \biggr)^{1/p}< 1 $$

holds for all $p\geq1$.

Proof

Let $p\in[1, \infty)$ and

$$ f_{1}(p)=\frac{1}{3p}+ \biggl(\frac{2}{e^{2}} \biggr)^{1/p}. $$

(2.1)

Then (2.1) leads to

$$\begin{aligned}& \lim_{p\rightarrow\infty}f_{1}(p)=1, \end{aligned}$$

(2.2)

$$\begin{aligned}& \begin{aligned}[b] f_{1}^{\prime}(p)&=\frac{2}{p^{2}}\log \biggl( \frac{\sqrt{2}e}{2} \biggr) \biggl[ \biggl(\frac{\sqrt{2}}{e} \biggr)^{2/p} -\frac{1}{6\log(\frac{\sqrt{2}e}{2} )} \biggr] \\ &\geq\frac{2}{p^{2}}\log \biggl(\frac{\sqrt{2}e}{2} \biggr) \biggl[ \biggl( \frac{\sqrt{2}}{e} \biggr)^{2} -\frac{1}{6\log(\frac{\sqrt{2}e}{2} )} \biggr] \\ &=\frac{12\log(\frac{\sqrt{2}e}{2} )-e^{2}}{3e^{2}p^{2}}.\end{aligned} \end{aligned}$$

(2.3)

Note that

$$ 12\log \biggl(\frac{\sqrt{2}e}{2} \biggr)-e^{2}\simeq0.4521>0. $$

(2.4)

Therefore, Lemma 2.2 follows easily from (2.1)–(2.4). □

Lemma 2.3

The function

$$ f_{2}(x)=\frac{4(x^{2}+1)\arctan(x)+x(x^{2}+2)}{x(3x^{2}+2)} $$

(2.5)

is strictly decreasing from $(0, \infty)$on $(1/3, 3)$.

Proof

It follows from (2.5) that

$$ f_{2} \bigl(0^{+} \bigr)=3, \qquad\lim_{x\rightarrow\infty}f_{2}(x)= \frac{1}{3}, $$

(2.6)

where and in what follows $f (\lambda^{+} )$ denotes the right limit of the function f at λ.

Let

$$ \varphi_{1}(x)=4\arctan(x)+\frac{x(x^{2}+2)}{x^{2}+1}, \qquad \varphi_{2}(x)=\frac{x(3x^{2}+2)}{x^{2}+1}. $$

Then we clearly see that

$$\begin{aligned}& \varphi_{1} \bigl(0^{+} \bigr)=\varphi_{2} \bigl(0^{+} \bigr)=0, \qquad f_{2}(x)=\frac{\varphi_{1}(x)}{\varphi_{2}(x)}, \\& \frac{\varphi_{1}^{\prime}(x)}{\varphi_{2}^{\prime}(x)}=\frac {x^{2}+3}{3x^{2}+1}. \end{aligned}$$

(2.7)

It is not difficult to verify that the function $x\rightarrow\varphi _{1}^{\prime}(x)/\varphi_{2}^{\prime}(x)$ is strictly decreasing on $(0, \infty)$.

Therefore, Lemma 2.3 follows from (2.6), (2.7), and Lemma 2.1 together with the monotonicity of the function $\varphi_{1}^{\prime}(x)/\varphi _{2}^{\prime}(x)$ on the interval $(0, \infty)$. □

Lemma 2.4

Let $0< u<1$, $p\geq1$, and

$$ g(u, p; x)=\frac{p}{2}\log \biggl(\frac{(1-u)x^{2}+2}{x^{2}+2} \biggr)+ \frac{1}{2}\log \biggl(\frac{x^{2}+2}{2(x^{2}+1)} \biggr)-\frac{\arctan (x)}{x}+1. $$

(2.8)

Then the following statements are true:

• (1) $g(u, p; x)>0$for all $x\in(0, \infty)$if and only if $u\leq1/(3p)$;

• (2) $g(u, p; x)<0$for all $x\in(0, \infty)$if and only if $u\geq 1-(2/e^{2})^{1/p}$.

Proof

From (2.8) we clearly see that

$$\begin{aligned}& g \bigl(u, p; 0^{+} \bigr)=0, \end{aligned}$$

(2.9)

$$\begin{aligned}& \lim_{x\rightarrow\infty}g(u, p; x)=\frac{p}{2}\log(1-u)+1- \frac{1}{2}\log2. \end{aligned}$$

(2.10)

Let

$$ g_{0}(p, x)=\frac{(x^{2}+2)[(x^{2}+2)\arctan (x)-2x]}{x^{2}[(x^{2}+2)\arctan(x)+2(p-1)x]}. $$

(2.11)

Then

$$ g_{0} \bigl(p, 0^{+} \bigr)=\frac{1}{3p}, \qquad\lim _{x\rightarrow\infty}g_{0}(p, x)=1. $$

(2.12)

Differentiating $g(u, p; x)$ with respect to x gives

$$ \frac{\partial g(u, p; x)}{\partial x}=\frac{(x^{2}+2)\arctan (x)+2(p-1)x}{(x^{2}+2)[(1-u)x^{2}+2]} \bigl[g_{0}(p, x)-u \bigr]. $$

(2.13)

Let

$$ g_{1}(x)=\arctan(x)-\frac{2x}{x^{2}+2}, \qquad g_{2}(x)= \frac{x^{2}}{x^{2}+2} \biggl[\arctan(x)+\frac{2(p-1)x}{x^{2}+2} \biggr]. $$

Then we clearly see that

$$\begin{aligned}& g_{0}(p, x)=\frac{g_{1}(x)}{g_{2}(x)}, \qquad g_{1} \bigl(0^{+} \bigr)=g_{2} \bigl(0^{+} \bigr)=0, \end{aligned}$$

(2.14)

$$\begin{aligned}& \begin{aligned}[b] \frac{g_{1}^{\prime}(x)}{g_{2}^{\prime}(x)}&=\frac {x(x^{2}+2)(3x^{2}+2)}{4(x^{2}+1)(x^{2}+2)\arctan (x)+x[(3-2p)x^{4}+2(5p-3)x^{2}+4(3p-2)]} \\ &=\frac{1}{\frac{4(x^{2}+1)\arctan(x)+x(x^{2}+2)}{x(3x^{2}+2)}+\frac {2(p-1)}{3}\frac{23x^{2}+22}{(x^{2}+2)(3x^{2}+2)}-\frac{2(p-1)}{3}}.\end{aligned} \end{aligned}$$

(2.15)

It is not difficult to verify that the function $x\rightarrow (23x^{2}+22)/[(x^{2}+2)(3x^{2}+2)]$ is strictly decreasing on $(0, \infty)$. Then from Lemma 2.3 and (2.15) we know that $g_{1}^{\prime }(x)/g_{2}^{\prime}(x)$ is strictly increasing on $(0, \infty)$. Therefore, the fact that the function $x\rightarrow g_{0}(x, p)$ is strictly increasing on $(0, \infty)$ follows from Lemma 2.1 and (2.14) together with the monotonicity of $g_{1}^{\prime}(x)/g_{2}^{\prime}(x)$ on the interval $(0, \infty)$.

From Lemma 2.2 we know that the interval $(0, 1)$ can be expressed by

$$ (0, 1)= \biggl(0, \frac{1}{3p} \biggr]\cup \biggl(\frac{1}{3p}, 1- \biggl( \frac{2}{e^{2}} \biggr)^{1/p} \biggr)\cup\biggl[1- \biggl( \frac{2}{e^{2}} \biggr)^{1/p}, 1 \biggr). $$

We divide the proof into three cases.

Case 1: $0< u\leq1/(3p)$. Then (2.12) and (2.13) together with the monotonicity of the function $x\rightarrow g_{0}(x, p)$ lead to the conclusion that the function $x\rightarrow g(u, p; x)$ is strictly increasing on $(0, \infty)$. Therefore $g(u, p; x)>0$ for all $x\in(0, \infty)$ follows from (2.9) and the monotonicity of the function $x\rightarrow g(u, p; x)$ on the interval $(0, \infty)$.

Case 2: $1-(2/e^{2})^{1/p}\leq u<1$. Then from (2.10), (2.12), (2.13), Lemma 2.2, and the monotonicity of the function $x\rightarrow g_{0}(x, p)$, we clearly see that

$$ \lim_{x\rightarrow\infty}g(u, p; x)\leq0, $$

(2.16)

and there exists $x_{0}\in(0, \infty)$ such that the function $x\rightarrow g(u, p; x)$ is strictly decreasing on $(0, x_{0})$ and strictly increasing on $(x_{0}, \infty)$. Therefore $g(u, p; x)<0$ for all $x\in(0, \infty)$ follows from (2.9) and (2.16) together with the piecewise monotonicity of the function $x\rightarrow g(u, p; x)$ on the interval $(0, \infty)$.

Case 3: $1/(3p)< u<1-(2/e^{2})^{1/p}$. Then it follows from (2.10), (2.12), (2.13), and the monotonicity of the function $x\rightarrow g_{0}(x, p)$ that

$$ \lim_{x\rightarrow\infty}g(u, p; x)>0, $$

(2.17)

and there exists $x^{\ast}\in(0, \infty)$ such that the function $x\rightarrow g(u, p; x)$ is strictly decreasing on $(0, x^{\ast})$ and strictly increasing on $(x^{\ast}, \infty)$. Therefore, there exists $\tau\in(0, \infty)$ such that $g(u, p; x)<0$ for $x\in(0, \tau)$ and $g(u, p; x)>0$ for $x\in(\tau, \infty)$ follows from (2.9) and (2.17) together with the piecewise monotonicity of the function $x\rightarrow g(u, p; x)$ on the interval $(0, \infty)$. □

3 Main result

Theorem 3.1

Let $p\geq1$, $0<\lambda, \mu<1/2$, andσandτbe any two different positive real numbers. Then the double inequality

$$ \mathbf{GA}_{\lambda, p}(\sigma, \tau)< \mathbf{SY}(\sigma, \tau)< \mathbf{GA}_{\mu, p}(\sigma, \tau) $$

holds if and only if

$$ \lambda\leq\frac{1}{2}-\frac{1}{2}\sqrt{1- \biggl( \frac{2}{e^{2}} \biggr)^{1/p}}, \qquad\mu\geq\frac{1}{2}- \frac{\sqrt{3p}}{6p}. $$

Proof

From (1.1)–(1.3) we clearly see that both $\mathbf{GA}_{\theta, p}(\sigma, \tau)$ and $\mathbf{SY}(\sigma, \tau)$ are symmetric and homogenous of degree one with respect to their variables σ and τ. Without loss of generality, we assume that $\sigma>\tau>0$. Let $0<\theta<1/2$ and $x=(\sigma-\tau)/\sqrt{2\sigma\tau}>0$. Then (1.1)–(1.3) lead to

$$\begin{aligned}& \mathbf{SY}(\sigma, \tau)=\mathbf{G}(\sigma, \tau)\sqrt{1+x^{2}}e^{\frac {\arctan(x)}{x}-1}, \\& \mathbf{GA}_{\theta, p}(\sigma, \tau)=\mathbf{G}(\sigma, \tau)\sqrt{1+ \frac{x^{2}}{2}} \biggl[\frac{ (1-(1-2\theta)^{2} )x^{2}+2}{x^{2}+2} \biggr]^{p/2}, \\& \begin{aligned}[b] &\log \bigl[\mathbf{GA}_{\theta, p}(\sigma, \tau) \bigr]-\log \bigl[ \mathbf{SY}( \sigma, \tau) \bigr] \\ &\quad=\frac{p}{2}\log \biggl[\frac{ (1-(1-2\theta)^{2} )x^{2}+2}{x^{2}+2} \biggr] +\frac{1}{2} \log \biggl(\frac{x^{2}+2}{2(x^{2}+1)} \biggr)-\frac{\arctan(x)}{x}+1 \\ &\quad=g \bigl((1-2\theta)^{2}, p;x \bigr),\end{aligned} \end{aligned}$$

(3.1)

where $g(\cdot, p; x)$ is defined by (2.8).

Therefore, Theorem 3.1 follows easily from Lemma 2.4 and (3.1). □

4 Applications

Let $p=1, 2$. Then Theorem 3.1 leads to Theorem 4.1 immediately, which provides the sharp bounds for the Sándor–Yang mean in terms of the one-parameter geometric and harmonic means.

Theorem 4.1

Let $0<\lambda_{1}, \lambda_{2}, \mu_{1}, \mu_{2}<1/2$, andσandτbe any two distinct positive real numbers. Then the double inequalities

$$ \mathbf{G} \bigl[\lambda_{1}\sigma+(1-\lambda_{1})\tau, \lambda_{1}\tau+(1-\lambda_{1})\sigma \bigr]< \mathbf{SY}( \sigma, \tau)< \mathbf{G} \bigl[\mu_{1}\sigma+(1-\mu_{1}) \tau, \mu_{1}\tau+(1-\mu_{1})\sigma \bigr] $$

and

$$ \mathbf{H} \bigl[\lambda_{2}\sigma+(1-\lambda_{2})\tau, \lambda_{2}\tau+(1-\lambda_{2})\sigma \bigr]< \mathbf{SY}( \sigma, \tau)< \mathbf{H} \bigl[\mu_{2}\sigma+(1-\mu_{2})\tau, \mu_{2}\tau+(1-\mu_{2})\sigma \bigr] $$

hold if and only if

$$\begin{gathered} \lambda_{1}\leq\frac{1}{2}-\frac{1}{2}\sqrt{1- \frac{2}{e^{2}}}\simeq0.0730, \qquad\mu_{1}\geq\frac{1}{2}- \frac{\sqrt{3}}{6}\simeq0.2113, \\ \lambda_{2}\leq\frac{1}{2}-\frac{1}{2}\sqrt{1- \frac{\sqrt{2}}{e}}\simeq0.1537, \qquad\mu_{2}\geq\frac{1}{2}- \frac{\sqrt{6}}{12}\simeq0.2959.\end{gathered} $$

Theorem 3.1 and (1.2) also lead to Theorem 4.2, which gives the sharp bounds for the Yang mean in terms of the two-parameter geometric and arithmetic mean and the quadratic and geometric means.

Theorem 4.2

Let $p\geq1$, $0<\alpha, \beta<1/2$, andσandτbe any two different positive real numbers. Then the two-sided inequality

$$ \frac{\mathbf{G}(\sigma, \tau)}{\log[\mathbf{GA}_{\alpha, p}(\sigma, \tau)]-\log[\mathbf{Q}(\sigma, \tau)]+1}< \mathbf{U}(\sigma, \tau)< \frac {\mathbf{G}(\sigma, \tau)}{\log[\mathbf{GA}_{\beta, p}(\sigma, \tau )]-\log[\mathbf{Q}(\sigma, \tau)]+1} $$

takes place if and only if

$$ \alpha\geq\frac{1}{2}-\frac{\sqrt{3p}}{6p}, \qquad\beta\leq \frac{1}{2}-\frac{1}{2}\sqrt{1- \biggl(\frac{2}{e^{2}} \biggr)^{1/p}}. $$

Let $\sigma>\tau=1/2$, $\alpha=1/2-\sqrt{3p}/(6p)$, and $\beta=1/2-\sqrt {1-(2/e^{2})^{1/p}}/2$. Then it follows from (1.1), (1.3) that

$$\begin{aligned}& \mathbf{U} \biggl(\sigma, \frac{1}{2} \biggr)=\frac{2\sigma-1}{2\sqrt {2}\arctan(\frac{2\sigma-1}{2\sqrt{\sigma}} )}, \end{aligned}$$

(4.1)

$$\begin{aligned}& \begin{aligned}[b] &\mathbf{GA}_{1/2-\sqrt{3p}/(6p), p} \biggl(\sigma, \frac{1}{2} \biggr) \\ &\quad= \biggl[\frac{4(3p-1)\sigma^{2}+4(3p+1)\sigma+3p-1}{48p} \biggr]^{p} \biggl(\frac{2\sigma+1}{4} \biggr)^{1-p},\end{aligned} \end{aligned}$$

(4.2)

$$\begin{aligned}& \begin{aligned}[b] &\mathbf{GA}_{1/2-\sqrt{1-(2/e^{2})^{1/p}}/2, p} \biggl(\sigma, \frac {1}{2} \biggr) \\ &\quad= \biggl[\frac{4\times2^{1/p}\sigma^{2}+4 (2e^{2/p}-2^{1/p} )\sigma +2^{1/p}}{16e^{2/p}} \biggr]^{p} \biggl(\frac{2\sigma+1}{4} \biggr)^{1-p}.\end{aligned} \end{aligned}$$

(4.3)

From Theorem 4.2 and (4.1)–(4.3) we obtain Theorem 4.3, which presents new one-parameter bounds for the inverse tangent function.

Theorem 4.3

The double inequality

$$\begin{gathered} \frac{2\sigma-1}{2\sqrt{\sigma}} \biggl[p\log \bigl(4\times2^{1/p}\sigma^{2}+4 \bigl(2e^{2/p}-2^{1/p} \bigr)\sigma+2^{1/p} \bigr) \\ \quad\quad{}+(1-p)\log(2\sigma+1)-\frac{1}{2}\log \bigl(4\sigma^{2}+1 \bigr)-p\log4-1-\frac{1}{2}\log2 \biggr] \\ \quad< \arctan \biggl(\frac{2\sigma-1}{2\sqrt{\sigma}} \biggr)< \frac{2\sigma -1}{2\sqrt{\sigma}} \biggl[p\log \bigl(4(3p-1)\sigma^{2}+4(3p+1)\sigma+3p-1 \bigr) \\ \qquad{}+(1-p)\log(2\sigma+1)-\frac{1}{2}\log \bigl(4\sigma^{2}+1 \bigr)-p (\log p+\log3+2\log2 )+1-\frac{1}{2}\log2 \biggr]\end{gathered} $$

holds for all $\sigma>1/2$and $p\geq1$.

5 Consequences and discussion

In the article, we have given the sharp bounds for the Sándor–Yang mean

$$ \mathbf{SY}(\sigma, \tau)=\mathbf{Q}(\sigma, \tau)e^{\mathbf{G}(\sigma, \tau)/\mathbf{U}(\sigma, \tau)-1} $$

in terms of the two-parameter geometric and arithmetic mean

$$ \mathbf{GA}_{\eta, \nu}(\sigma, \tau)=\mathbf{G}^{\nu} \bigl[\eta \sigma+(1-\eta)\tau, \eta\tau+(1-\eta)\sigma \bigr]\mathbf{A}^{1-\nu}( \sigma, \tau) $$

and the one-parameter geometric and harmonic means

$$ \mathbf{G} \bigl[\lambda\sigma+(1-\lambda)\tau, \lambda\tau+(1-\lambda )\sigma \bigr] $$

and

$$ \mathbf{H} \bigl[\mu\sigma+(1-\mu)\tau, \mu\tau+(1-\mu)\sigma \bigr], $$

and have found the new bounds for the Yang mean

$$ \mathbf{U}(\sigma, \tau)=\frac{\sqrt{2}(\sigma-\tau)}{2\arctan(\frac {\sqrt{2}(\sigma-\tau)}{2\sqrt{\sigma\tau}} )} $$

and the inverse tangent function $\arctan[(2\sigma-1)/(2\sqrt{\sigma})]$.

6 Conclusion

In the article, we have proved that the double inequalities

$$ \mathbf{GA}_{\lambda, p}(\sigma, \tau)< \mathbf{SY}(\sigma, \tau)< \mathbf{GA}_{\mu, p}(\sigma, \tau) $$

and

$$ \frac{\mathbf{G}(\sigma, \tau)}{\log[\mathbf{GA}_{\mu, p}(\sigma, \tau )]-\log[\mathbf{Q}(\sigma, \tau)]+1}< \mathbf{U}(\sigma, \tau)< \frac {\mathbf{G}(\sigma, \tau)}{\log[\mathbf{GA}_{\lambda, p}(\sigma, \tau )]-\log[\mathbf{Q}(\sigma, \tau)]+1} $$

are valid for all distinct positive real numbers σ and τ if and only if

$$ \lambda\leq\frac{1}{2}-\frac{1}{2}\sqrt{1- \biggl( \frac{2}{e^{2}} \biggr)^{1/p}}, \qquad\mu\geq\frac{1}{2}- \frac{\sqrt{3p}}{6p}. $$

if $p\geq1$ and $\lambda, \mu\in(0, 1/2)$.