Open Access

Monotonicity and absolute monotonicity for the two-parameter hyperbolic and trigonometric functions with applications

Journal of Inequalities and Applications20162016:200

https://doi.org/10.1186/s13660-016-1143-8

Received: 10 May 2016

Accepted: 9 August 2016

Published: 17 August 2016

Abstract

In this paper, we present the monotonicity and absolute monotonicity properties for the two-parameter hyperbolic and trigonometric functions. As applications, we find several complete monotonicity properties for the functions involving the gamma function and provide the bounds for the error function.

Keywords

Stolarsky meanhyperbolic functiontrigonometric functiongamma functionerror functioncomplete monotonicityabsolute monotonicity

MSC

33B1033B1533B2026A4826D07

1 Introduction

Let \(p, q\in\mathbb{R}\) and \(a, b>0\) with \(a\neq b\). Then the Stolarsky mean \(S_{p, q}(a,b)\) [1] is given by
$$ S_{p, q}(a,b)= \textstyle\begin{cases} [\frac{q (a^{p}-b^{p} )}{p (a^{q}-b^{q} )} ]^{1/(p-q)}, &pq(p-q)\neq0, \\ {}[\frac{a^{p}-b^{p}}{p(\log a-\log b)} ]^{1/p}, &p\neq0, q=0, \\ {}[\frac{a^{q}-b^{q}}{q(\log a-\log b)} ]^{1/q}, &p=0, q\neq 0, \\ \exp [\frac{a^{p}\log a-b^{p}\log b}{a^{p}-b^{p}}-\frac{1}{p} ], &p=q\neq0, \\ \sqrt{ab}, &p=q=0. \end{cases} $$

It is well known that \(S_{p, q}(a,b)\) is continuous and symmetric on the domain \(\{(p, q, a, b): p, q\in\mathbb{R}, a>0, b>0\}\) and strictly increasing with respect to its parameters \(p, q\in \mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\). Many bivariate means are particular cases of the Stolarksy mean, and many remarkable inequalities and properties for this mean can be found in the literature [213]. We clearly see that the value \(S_{p,q}(a,b)\) in the case of \(pq(p-q)=0\) is the limit of the case of \(pq(p-q)\neq0\).

Let \(b>a>0\) and \(t=\log\sqrt{b/a}\in(0, \infty)\). Then the Stolarsky mean \(S_{p, q}(a,b)\) can be expressed by a hyperbolic function as follows:
$$ S_{p, q}(a,b)=\sqrt{ab}H_{p, q}(t), $$
(1.1)
where
$$ H_{p, q}(t)= \textstyle\begin{cases} (\frac{q\sinh(pt)}{p\sinh(qt)} )^{1/(p-q)}, &pq(p-q)\neq0, \\ (\frac{\sinh(pt)}{pt} )^{1/p}, &p\neq0, q=0, \\ (\frac{\sinh(qt)}{qt} )^{1/q}, &p=0, q\neq0, \\ \exp (t\coth(pt)-\frac{1}{p} ), &p=q\neq0, \\ 1, &p=q=0, \end{cases} $$
(1.2)
is the two-parameter hyperbolic sine function [14].
Let \(p, q\in[-2, 2]\) and \(t\in(0, \pi/2)\). Then the two-parameter trigonometric sine function \(T_{p, q}(t)\) [14] is given by
$$ T_{p, q}(t)= \textstyle\begin{cases} (\frac{q\sin(pt)}{p\sin(qt)} )^{1/(p-q)}, &pq(p-q)\neq0, \\ (\frac{\sin(pt)}{pt} )^{1/p}, &p\neq0, q=0, \\ (\frac{\sin(qt)}{qt} )^{1/q}, &p=0, q\neq0, \\ \exp (t\cot(pt)-\frac{1}{p} ), &p=q\neq0, \\ 1, &p=q=0. \end{cases} $$
(1.3)

The main purpose of this paper is to deal with the monotonicity of the functions \(t\mapsto[\log H_{p, q}(t)]/t\) and \(t\mapsto[\log H_{p, q}(t)]/t^{2}\) on the interval \((0, \infty)\) and with the absolute monotonicity of the functions \(t\mapsto\log T_{p, q}(t)\), \(t\mapsto [\log T_{p, q}(t)]/t\) and \(t\mapsto[\log T_{p, q}(t)]/t^{2}\) on the interval \((0, \pi/2)\). As applications, we shall present several complete monotonicity properties for the functions involving the gamma function and provide bounds for the error function.

2 Main results

Theorem 2.1

Let \(p, q\in\mathbb{R}\), \(t>0\), and \(H_{p, q}(t)\) be defined by (1.2). Then the function \(t\mapsto[\log H_{p, q}(t)]/t\) is strictly increasing (decreasing) and strictly concave (convex) from \((0, \infty)\) onto \((0, (p+q)/(|p|+|q|))\) (\(((p+q)/(|p|+|q|), 0)\)) if \(p+q>0\) (<0).

Proof

We only prove the desired result in the case of \(pq(p+q)\neq0\); the other cases can be derived easily from the continuity and limit values. Let
$$\begin{aligned}& f_{1}(t)=t \biggl[\frac{p\cosh(pt)}{\sinh(pt)}-\frac{q\cosh(qt)}{\sinh (qt)} \biggr]-\log \sinh\bigl(\vert p\vert t\bigr) \\& \hphantom{f_{1}(t)={}}{}+\log\sinh\bigl(\vert q\vert t\bigr)+\log \vert p\vert -\log{ \vert q\vert }, \\& f_{2}(t)=tf^{\prime}_{1}(t)-2f_{1}(t), \\& F_{1}(u)=\frac{u}{\sinh(u)},\qquad F_{2}(u)= \frac{u^{3}\cosh(u)}{\sinh^{3}(u)}. \end{aligned}$$
Then elaborated computations lead to
$$\begin{aligned}& f_{1}\bigl(0^{+}\bigr)=f_{2}\bigl(0^{+} \bigr)=\lim_{t\rightarrow0^{+}}\frac{\log H_{p, q}(t)}{t}=0, \end{aligned}$$
(2.1)
$$\begin{aligned}& \log H_{p,q}(t)= \frac{1}{p-q}\log \biggl(\frac{q\sinh(pt)}{p\sinh(qt)} \biggr) =\frac{1}{p-q}\log \biggl(\frac{|q|\sinh(|p|t)}{|p|\sinh(|q|t)} \biggr) \\& \hphantom{\log H_{p,q}(t)}= \frac{|p|-|q|}{p-q}t+\frac{1}{p-q}\log \biggl[ \frac{|q| (1-e^{-2|p|t} )}{|p| (1-e^{-2|q|t} )} \biggr], \\& \lim_{t\rightarrow\infty}\frac{\log H_{p, q}(t)}{t}=\frac{|p|-|q|}{p-q}= \frac{p+q}{|p|+|q|}, \end{aligned}$$
(2.2)
$$\begin{aligned}& \biggl[\frac{\log H_{p, q}(t)}{t} \biggr]^{\prime}=\frac{f_{1}(t)}{(p-q)t^{2}}= \frac {p+q}{(|p|+|q|)t^{2}}\times\frac{f_{1}(t)}{|p|-|q|}, \end{aligned}$$
(2.3)
$$\begin{aligned}& f^{\prime}_{1}(t)=\frac{1}{t} \biggl[\frac{(qt)^{2}}{\sinh^{2}(qt)}- \frac {(pt)^{2}}{\sinh^{2}(pt)} \biggr], \\& \frac{f^{\prime}_{1}(t)}{|p|-|q|}= \frac {F^{2}_{1}(qt)-F^{2}_{1}(pt)}{(|p|-|q|)t}=\frac {F^{2}_{1}(|qt|)-F^{2}_{1}(|pt|)}{|pt|-|qt|} \\& \hphantom{\frac{f^{\prime}_{1}(t)}{|p|-|q|}} = -\bigl[F_{1}\bigl(|qt|\bigr)+F_{1}\bigl(|pt|\bigr)\bigr] \frac{F_{1}(|qt|)-F_{1}(|pt|)}{|qt|-|pt|}, \end{aligned}$$
(2.4)
$$\begin{aligned}& \biggl[\frac{\log H_{p, q}(t)}{t} \biggr]^{\prime\prime}=\frac{f_{2}(t)}{(p-q)t^{3}}= \frac {p+q}{(|p|+|q|)t^{3}}\times\frac{f_{2}(t)}{|p|-|q|}, \end{aligned}$$
(2.5)
$$\begin{aligned}& f^{\prime}_{2}(t)=\frac{2}{t} \biggl[\frac{(pt)^{3}\cosh(pt)}{\sinh ^{3}(pt)}- \frac{(qt)^{3}\cosh(qt)}{\sinh^{3}(qt)} \biggr], \\& \frac{f^{\prime}_{2}(t)}{|p|-|q|}=\frac{2[F_{2}(|pt|)-F_{2}(|qt|)]}{|pt|-|qt|}, \end{aligned}$$
(2.6)
$$\begin{aligned}& F^{\prime}_{1}(u)=-\frac{\cosh(u)}{\sinh^{2}(u)}\bigl[u-\tanh(u) \bigr]< 0, \end{aligned}$$
(2.7)
$$\begin{aligned}& F^{\prime}_{2}(u)=-\frac{3u^{3}}{\sinh^{4}(u)} \biggl[\frac{\sinh (2u)}{2u}- \frac{2+\cosh(2u)}{3} \biggr]< 0 \end{aligned}$$
(2.8)
for \(u>0\), where the inequality in (2.8) is the Cusa-type inequality given in [15].
It follows from (2.1), (2.4), and (2.6)-(2.8) that
$$ \frac{f_{1}(t)}{|p|-|q|}>0 $$
(2.9)
and
$$ \frac{f_{2}(t)}{|p|-|q|}< 0 $$
(2.10)
for \(t\in(0, \infty)\).

Therefore, Theorem 2.1 follows easily from (2.1)-(2.3), (2.5), (2.9), and (2.10). □

Theorem 2.2

Let \(p, q\in\mathbb{R}\) and \(t>0\), and let \(H_{p, q}(t)\) be defined by (1.2). Then the function \(t\mapsto[\log H_{p, q}(t)]/t^{2}\) is strictly decreasing (increasing) from \((0, \infty)\) onto \((0, (p+q)/6)\) (\(((p+q)/6, 0)\)) if \(p+q>0\) (<0).

Proof

Let \(g_{1}(t)=[\log H_{p, q}(t)]/t\) and \(g_{2}(t)=t\). Then we clearly see that
$$ \frac{g^{\prime}_{1}(t)}{g^{\prime}_{2}(t)}=g^{\prime}_{1}(t)= \biggl[ \frac{\log H_{p, q}(t)}{t} \biggr]^{\prime}, $$
(2.11)
and (2.1) leads
$$ \frac{\log H_{p, q}(t)}{t^{2}}=\frac{g_{1}(t)}{g_{2}(t)}=\frac {g_{1}(t)-g_{1}(0^{+})}{g_{2}(t)-g_{2}(0^{+})}. $$
(2.12)

From Theorem 2.1, (2.11), (2.12), and the well-known monotone form of l’Hôpital’s rule [16] we know that the function \(t\mapsto[\log H_{p, q}(t)]/t^{2}\) is strictly decreasing (increasing) on \((0, \infty)\) if \(p+q>0\) (<0).

It follows from l’Hôpital’s rule and (2.2) that
$$ \lim_{t\rightarrow0^{+}}\frac{\log H_{p, q}(t)}{t^{2}}=\frac{p+q}{6} \quad \mbox{and}\quad \lim_{t\rightarrow \infty}\frac{\log H_{p, q}(t)}{t^{2}}=0. $$
 □

From (1.1) and Theorem 2.2 we get the following corollary.

Corollary 2.1

For \(a, b>0\) with \(a\neq b\), we have the double inequality
$$ \sqrt{ab}< (>)\, S_{p, q}(a,b)< (>)\, \sqrt{ab} e^{\frac{p+q}{24}(\log b-\log a)^{2}} $$
if \(p+q>0\) (<0).

Letting \(b>a>0\), \(t=\log\sqrt{b/a}>0\), and \((p,q)=(1,0),(1,1),(3/2,1/2)\) in Corollary 2.1, we get the following corollary.

Corollary 2.2

We have the inequalities
$$ \frac{\sinh(t)}{t}< e^{t^{2}/6},\qquad e^{t\cosh(t)-1}< e^{t^{2}/3}, \qquad \frac{2\cosh(t)+1}{3}< e^{t^{2}/3} $$
for all \(t>0\).
Next, we recall the definition of absolutely monotonic function [17]. A real-valued function f is said to be absolutely monotonic on the interval I if f has derivatives of all orders on I and
$$ f^{(n)}(x)>0 $$
for all \(x\in I\) and \(n\geq0\).

Theorem 2.3

Let \(p, q\in[-2, 2]\) and \(t\in(0, \pi/2)\), and let \(T_{p, q}(t)\) be defined by (1.3). Then the functions \(t\rightarrow\log T_{p, q}(t)\), \(t\rightarrow[\log T_{p, q}(t)]/t\), and \(t\rightarrow[\log T_{p, q}(t)]/t^{2}\) are absolutely monotonic on \((0, \pi/2)\) if \(p+q<0\). Moreover, the functions \(t\rightarrow-\log T_{p, q}(t)\), \(t\rightarrow-[\log T_{p,q}(t)]/t\), and \(t\rightarrow-[\log T_{p, q}(t)]/t^{2}\) are absolutely monotonic on \((0,\pi/2)\) if \(p+q>0\).

Proof

We only prove the desired result in the case of \(pq(p+q)\neq0\); the other cases can be derived easily from the continuity and limit values.

Let \(i=0,1,2\). Then from (1.3) and the power series formula
$$ \log\frac{\sin(t)}{t}=-\sum_{n=1}^{\infty} \frac{2^{2n-1}|B_{2n}|}{n(2n)!}t^{2n},\quad |t|< \pi, $$
listed in [18], 4.3.71, we get
$$\begin{aligned}& \log T_{p, q}(t)=\frac{1}{p-q}\log \biggl( \frac{q\sin(pt)}{p\sin(qt)} \biggr)=\frac {1}{p-q}\log \biggl(\frac{|qt|\sin(|pt|)}{|pt|\sin(|qt|)} \biggr) \\& \hphantom{\log T_{p, q}(t)}=-(p+q)t^{2}\sum_{n=1}^{\infty} \frac{2^{2n}|B_{2n}| (p^{2n}-q^{2n} )}{2n(2n)! (p^{2}-q^{2} )}t^{2n-2}, \\& \frac{\log T_{p, q}(t)}{t^{i}}=-(p+q)t^{2-i}\sum_{n=1}^{\infty} \frac{2^{2n}|B_{2n}| (p^{2n}-q^{2n} )}{2n(2n)! (p^{2}-q^{2} )}t^{2n-2}, \end{aligned}$$
(2.13)
where \(B_{n}\) are the Bernoulli numbers.

Therefore, Theorem 2.3 follows easily from (2.13). □

Let \((p,q)=(1,0), (1,1), (3/2,1/2)\) in Theorem 2.3. Then we immediately get the following corollary.

Corollary 2.3

We have the inequalities
$$\begin{aligned}& \biggl(\frac{2}{\pi} \biggr)^{4t^{2}/\pi^{2}}< \frac{\sin (t)}{t}< e^{-t^{2}/6}, \end{aligned}$$
(2.14)
$$\begin{aligned}& 1-\frac{4t^{2}}{\pi^{2}}< \frac{t}{\tan(t)}< 1-\frac{t^{2}}{3}, \\& 3^{-4t^{2}/\pi^{2}}< \frac{2\cos(t)+1}{3}< e^{-t^{2}/3} \end{aligned}$$
(2.15)
for all \(t\in(0,\pi/2)\).

Remark 2.1

The second inequality in (2.14) was first proved by Yang [19], and the double inequality (2.15) can be found in [20], which is better than the Redheffer-type inequality in Theorem 3 of [21].

Remark 2.2

Bhayo and Sándor [22], equation (3.3), presented the double inequality
$$ 1-\frac{4t^{2}}{\pi^{2}}< \frac{t}{\tan(t)}< \frac{\pi^{2}}{8}- \frac{t^{2}}{2} $$
(2.16)
for all \(t\in(0, \pi/2)\). The second inequality in (2.16) is better than the second inequality in (2.15) for \(t\in(\sqrt{3\pi^{2}/4-6},\pi/2)\).

3 Applications

Recall that a real-valued function f is said to be completely monotonic [23] on the interval I if f has derivatives of all order on I and
$$ (-1)^{n}f^{(n)}(x)\geq0 $$
for all \(n\geq0\) and \(x\in I\). The set of all completely monotonic functions on I is denoted by \(\operatorname{CM}[I]\). A positive function f is said to be logarithmically completely monotonic on the interval I if its logarithm logf is completely monotonic on I. The class of all logarithmically completely monotonic functions on I is denoted by \(\operatorname{LCM}[I]\). The famous Bernstein theorem [17] implies that the function
$$ f(x)= \int_{0}^{\infty}e^{-xt}g(t)\, dt $$
is completely monotonic on \((0, \infty)\) if and only if \(g(t)\geq0\) for all \(t\in(0,\infty)\) if \(g(t)\) is continuous on \((0, \infty)\).

Theorem 3.1

Let \(s, t, r\in\mathbb{R}\), \(\rho=\min\{s, t, r\}\), \(x\in(-\rho, \infty)\), let \(\Gamma(u)=\int_{0}^{\infty}e^{-t}t^{u-1}\,dt\) (\(u>0\)) be the gamma function, \(\psi(u)=\Gamma^{\prime}(u)/\Gamma(u)\) be the psi function, and the function \(x\rightarrow v(s, t, r; x)\) be defined by
$$ v(s, t, r; x)= \textstyle\begin{cases} e^{-\psi(x+r)} [\frac{\Gamma(x+t)}{\Gamma(x+s)} ]^{1/(t-s)}, &t\neq s, \\ e^{-\psi(x+r)}\lim_{t\rightarrow s} [\frac{\Gamma(x+t)}{\Gamma (x+s)} ]^{1/(t-s)}=e^{\psi(x+s)-\psi(x+r)}, &t=s. \end{cases} $$
(3.1)
Then \(v(s, t, r; x)\in \operatorname{LCM}[(-\rho, \infty)]\) if and only if \(r\leq \min\{s, t\}\), and \(1/v(s, t, r; x)\in \operatorname{LCM}[(-\rho, \infty)]\) if and only if \(r\geq(s+t)/2\).

Proof

We only prove the desired result in the case of \(t\neq s\) because the case of \(t=s\) can be derived easily from the continuity and limit values.

Let \(L(a,b)=(b-a)/(\log b-\log a)\) be the logarithmic mean of two distinct positive real numbers a and b, \(u>0\), \(y=|(t-s)u/2|\), and \(p(s, t, r; u)\) and \(q(s,t,r;u)\) be respectively defined by
$$\begin{aligned}& p(s,t,r;u)=\frac{\log e^{(\rho-r)u}-\log\frac{e^{(\rho-s)u}-e^{(\rho-t)u}}{(t-s)u}}{u}, \\& q(s, t, r; u)=\frac{e^{(\rho-r)u}-\frac{e^{(\rho-s)u}-e^{(\rho-t)u}}{(t-s)u}}{1-e^{-u}}. \end{aligned}$$
Then we clearly see that
$$\begin{aligned}& p(s,t,r;u)=-r-\frac{1}{u}\log\frac{e^{-su}-e^{-tu}}{(t-s)u} =-r+\frac{t+s}{2}- \frac{|t-s|}{2} \biggl(\frac{1}{y}\log\frac{\sinh (y)}{y} \biggr), \end{aligned}$$
(3.2)
$$\begin{aligned}& q(s, t, r; u)=\frac{u}{1-e^{-u}}L \biggl(e^{(\rho-r)u}, \frac{e^{(\rho-s)u}-e^{(\rho-t)u}}{(t-s)u} \biggr)p(s,t,r; u). \end{aligned}$$
(3.3)
It follows from (1.2) and Theorem 2.1 that the function \(y\rightarrow[\log(\sinh(y)/y)]/y\) is strictly increasing from \((0, \infty)\) onto \((0, 1)\). Then (3.2) leads to the conclusion that
$$ \min\{s, t\}-r=-r+\frac{t+s}{2}-\frac{|t-s|}{2}< p(s, t, r; u)< -r+ \frac{t+s}{2}. $$
Therefore,
$$ p(s, t, r; u)\geq0 $$
(3.4)
for all \(u>0\) if and only if \(r\leq\min\{s, t\}\), and
$$ p(s, t, r; u)\leq0 $$
(3.5)
for all \(u>0\) if and only if \(r\geq(s+t)/2\).
From (3.1) and the integral formulas
$$\begin{aligned}& \log\Gamma(x)= \int_{0}^{\infty}\frac{1}{u} \biggl((x-1)e^{-u}-\frac {e^{-u}-e^{-xu}}{1-e^{-u}} \biggr)\,du, \\& \psi(x)= \int_{0}^{\infty} \biggl(\frac{e^{-u}}{u}- \frac {e^{-xu}}{1-e^{-u}} \biggr)\,du, \end{aligned}$$
given in [18], 6.1.50, 6.3.21, we get
$$\begin{aligned} \log v(s,t,r;x)&=\frac{\log\Gamma(x+t)-\log\Gamma(x+x)}{t-s}-\psi (x+r) \\ &= \int_{0}^{\infty}\frac{e^{-xu}}{1-e^{-u}} \biggl[ \frac {e^{-tu}-e^{-su}}{(t-s)u}+e^{-ru} \biggr]\,du \\ &= \int_{0}^{\infty}e^{-(x+\rho)u}q(s, t, r; u)\,du. \end{aligned}$$
(3.6)

Therefore, Theorem 3.1 follows easily from (3.3)-(3.6) and the Bernstein theorem. □

Remark 3.1

Qi and Guo [24] gave a sufficient condition for \(v(s, t, r; x)\in \operatorname{LCM}[(-\rho, \infty)]\) and a necessary and sufficient condition for \(1/v(s, t, r; x)\in \operatorname{LCM}[(-\rho, \infty)]\) by using different methods.

Theorem 3.2

Let \(a, b, c\in\mathbb{R}\), \(\rho=\min\{a, b, c\}\), \(x\in(-\rho, \infty)\), and let the function \(x\rightarrow U(a, b, c; x)\) be defined by
$$ U(a, b, c; x)= \textstyle\begin{cases} \frac{1}{x+c} (\frac{\Gamma(x+a)}{\Gamma(x+b)} )^{1/(a-b)},& b\neq a, \\ \lim_{b\rightarrow a}\frac{1}{x+c} (\frac{\Gamma(x+a)}{\Gamma (x+b)} )^{1/(a-b)}=\frac{1}{x+c}e^{\psi(x+a)},& b=a. \end{cases} $$
(3.7)
Then \(U(a, b, c; x)\in \operatorname{LCM}[(-\rho, \infty)]\) if and only if \(c\leq(a+b-\max\{|a-b|, 1\})/2\), and \(1/U(a, b, c; x)\in \operatorname{LCM}[(-\rho, \infty)]\) if and only if \(c\geq(a+b-\min\{|a-b|, 1\})/2\).

Proof

We only prove the desired result in the case of \(b\neq a\) because the case of \(b=a\) can be derived easily from the continuity and limit values.

We clearly see that \(U(a, b, c; x)\in \operatorname{LCM}[(-\rho, \infty)]\) if and only if \(-[\log U(a, b, c; x)]^{\prime}\in \operatorname{CM}[(-\rho, \infty)]\) and that \(1/U(a, b, c; x)\in \operatorname{LCM}[(-\rho, \infty)]\) if and only if \([\log U(a, b, c; x)]^{\prime}\in \operatorname{CM}[(-\rho, \infty)]\).

Let \(t>0\), \(H_{p,q}(t)\) be defined by (1.2), and \(p(a, b, c; t)\) and \(q(a, b, c; t)\) be respectively defined by
$$ p(a, b, c; t)=\frac{\log e^{(\rho-c)t}-\log\frac{e^{(\rho-a)t}-e^{(\rho-b)t}}{(b-a) (1-e^{-t} )}}{t} $$
and
$$ q(a, b, c; t)=e^{(\rho-c)t}-\frac{e^{(\rho-a)t}-e^{(\rho-b)t}}{(b-a) (1-e^{-t} )}. $$
Then we clearly see that
$$\begin{aligned} p(a,b,c;t)=&-c-\frac{1}{t}\log\frac{e^{-at}-e^{-bt}}{(b-a) (1-e^{-t} )} \\ =&\frac{a+b-1}{2}-c-\frac{1}{t}\log \biggl[\frac{\sinh (\vert \frac {(b-a)t}{2}\vert )}{|b-a|\sinh (\frac{t}{2} )} \biggr] \\ =&\frac{a+b-2c-1}{2}-\frac{|b-a|-1}{2}\frac{\log H_{|b-a|,1}(t/2)}{t/2} \end{aligned}$$
(3.8)
and
$$ q(a, b, c; t)=tL \biggl(e^{(\rho-c)t}, \frac{e^{(\rho-a)t}-e^{(\rho-b)t}}{(b-a) (1-e^{-t} )} \biggr)p(a, b, c; t). $$
(3.9)
It follows from Theorem 2.1 and (3.8) that the function \(t\rightarrow p(a, b, c; t)\) is strictly monotonic on \((0, \infty)\) and
$$ p\bigl(a, b, c; 0^{+}\bigr)=\frac{a+b-2c}{2},\qquad p(a, b, c; \infty)=\frac{a+b-2c}{2}-\frac{|b-a|-1}{2}. $$
(3.10)
The monotonicity of the function \(t\rightarrow p(a, b, c; t)\) on the interval \((0, \infty)\) and (3.10) lead to the conclusion that
$$ p(a, b, c;t)\geq(\leq)\, 0 $$
(3.11)
for all \(t\in(0, \infty)\) if and only if \(\min(\max)\{p(a, b, c; 0^{+}), p(a, b, c; \infty)\}\geq(\leq)\, 0\), that is, \(c\leq (\geq)\,(a+b-\max(\min)\{|a-b|, 1\})/2\).
From (3.7) and the formulas
$$ \psi(x)= \int_{0}^{\infty} \biggl(\frac{e^{-t}}{t}- \frac {e^{-xt}}{1-e^{-t}} \biggr)\,dt,\qquad \frac{1}{x}= \int_{0}^{\infty}e^{-xt}\,dt $$
we have
$$\begin{aligned} - \bigl(\log U(a, b, c; x) \bigr)^{\prime}&=\frac{1}{x+c}- \frac{\psi(x+b)-\psi (x+a)}{b-a} \\ &= \int_{0}^{\infty}e^{-(x+c)t}\,dt- \int_{0}^{\infty}\frac {e^{-(x+a)t}-e^{-(x+b)t}}{(b-a) (1-e^{-t} )}\,dt \\ &= \int_{0}^{\infty}e^{-(x+\rho)t}q(a, b, c; t)\,dt. \end{aligned}$$
(3.12)

Therefore, Theorem 3.2 follows from (3.9), (3.11), (3.12), and the Bernstein theorem. □

Remark 3.2

Qi [25] presented a sufficient condition for \(U(a, b, c; x)\in \operatorname{LCM}[(-\rho, \infty)]\) or \(1/U(a, b, c; x)\in \operatorname{LCM}[(-\rho, \infty)]\).

Theorem 3.3

Let \(\operatorname {erf}(x)=2\int_{0}^{x}e^{-t^{2}}\,dt/\sqrt{\pi}\) be the error function. Then we have the double inequality
$$ \frac{4}{\sqrt{\pi}}\arctan\frac{2e^{\sqrt{3}x}+1}{\sqrt{3}}+1-2\sqrt {\pi}< \operatorname {erf}(x) < \frac{4}{\sqrt{\pi}}\arctan\frac{2e^{\sqrt{3}x}+1}{\sqrt{3}}-\frac {4\sqrt{\pi}}{3} $$
for all \(x>0\).

Proof

It follows from the third inequality in Corollary 2.2 that
$$ e^{-u^{2}}-\frac{3}{2\cosh(\sqrt{3}u)+1}< 0 $$
(3.13)
for \(u>0\).
Let
$$\begin{aligned} F(x)&=\frac{2}{\sqrt{\pi}} \int_{0}^{x} \biggl(e^{-u^{2}}- \frac{3}{2\cosh (\sqrt{3}u)+1} \biggr)\,du \\ &=\operatorname {erf}(x)-\frac{6}{\sqrt{\pi}} \int_{0}^{x}\frac{1}{2\cosh(\sqrt{3}u)+1}\,du. \end{aligned}$$
(3.14)
Then
$$ F(0)=0, \qquad F(\infty)=1-\frac{2\sqrt{\pi}}{3}. $$
(3.15)
It follows from (3.13)-(3.15) that
$$ \frac{6}{\sqrt{\pi}} \int_{0}^{x}\frac{1}{2\cosh(\sqrt{3}u)+1}\,du+1- \frac {2\sqrt{\pi}}{3}< \operatorname {erf}(x)< \frac{6}{\sqrt{\pi}} \int_{0}^{x}\frac{1}{2\cosh(\sqrt{3}u)+1}\,du $$
(3.16)
for \(x>0\).

Therefore, Theorem 3.3 follows easily from (3.16). □

Declarations

Acknowledgements

The research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11401191 and by the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

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)
School of Mathematics and Computation Sciences, Hunan City University
(2)
Customer Service Center, State Grid Zhejiang Electric Power Research Institute

References

  1. Stolarsky, KB: Generalizations of the logarithmic mean. Math. Mag. 48, 87-92 (1975) MathSciNetView ArticleMATHGoogle Scholar
  2. Leach, EB, Sholander, MC: Extended mean values. Am. Math. Mon. 85(2), 84-90 (1978) MathSciNetView ArticleMATHGoogle Scholar
  3. Leach, EB, Sholander, MC: Extended mean values II. J. Math. Anal. Appl. 92(1), 207-223 (1983) MathSciNetView ArticleMATHGoogle Scholar
  4. Qi, F: Logarithmic convexity of the extended mean values. Proc. Am. Math. Soc. 130(6), 1787-1796 (2002) MathSciNetView ArticleMATHGoogle Scholar
  5. Yang, Z-H: On the homogeneous functions with two parameters and its monotonicity. JIPAM. J. Inequal. Pure Appl. Math. 6(4), Article ID 101 (2005) MathSciNetMATHGoogle Scholar
  6. Yang, Z-H: On the log-convexity of two-parameter homogeneous functions. Math. Inequal. Appl. 10(3), 499-516 (2007) MathSciNetMATHGoogle Scholar
  7. Chu, Y-M, Zhang, X-M: Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave. J. Math. Kyoto Univ. 48(1), 229-238 (2008) MathSciNetMATHGoogle Scholar
  8. Yang, Z-H: On the monotonicity and log-convexity of a four-parameter homogeneous mean. J. Inequal. Appl. 2008, Article ID 149286 (2008) MathSciNetView ArticleMATHGoogle Scholar
  9. Chu, Y-M, Zhang, X-M, Wang, G-D: The Schur geometrical convexity of the extended mean values. J. Convex Anal. 15(4), 707-718 (2008) MathSciNetMATHGoogle Scholar
  10. Witkowsik, A: Comparison theorem for two-parameter means. Math. Inequal. Appl. 12(1), 11-20 (2009) MathSciNetGoogle Scholar
  11. Yang, Z-H: Hölder, Chebyshev and Minkowski type inequality for Stolarsky means. Int. J. Math. Anal. 4(33-36), 1687-1696 (2010) MATHGoogle Scholar
  12. Xia, W-F, Wang, G-D, Chu, Y-M: Necessary and sufficient conditions for the Schur harmonic convexity or concavity of the extended means. Rev. Unión Mat. Argent. 52(1), 121-132 (2011) MathSciNetMATHGoogle Scholar
  13. Yang, Z-H, Chu, Y-M, Zhang, W: Accurate approximations for the complete elliptic integral of the second kind. J. Math. Anal. Appl. 438(2), 875-888 (2016) MathSciNetView ArticleMATHGoogle Scholar
  14. Yang, Z-H: Three families of two-parameter means constructed by trigonometric functions. J. Inequal. Appl. 2013, Article ID 541 (2013) MathSciNetView ArticleMATHGoogle Scholar
  15. Neuman, E, Sándor, J: On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities. Math. Inequal. Appl. 13(4), 715-723 (2010) MathSciNetMATHGoogle Scholar
  16. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Monotonicity rules in calculus. Am. Math. Mon. 113(9), 805-816 (2006) MathSciNetView ArticleMATHGoogle Scholar
  17. Widder, DV: The Laplace Transform. Princeton University Press, Princeton (1941) MATHGoogle Scholar
  18. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1970) MATHGoogle Scholar
  19. Yang, Z-H: Refinements of a two-sided inequality for trigonometric functions. J. Math. Inequal. 7(4), 601-615 (2013) MathSciNetView ArticleMATHGoogle Scholar
  20. Yang, Z-H, Jiang, Y-L, Song, Y-Q, Chu, Y-M: Sharp inequalities for trigonometric functions. Abstr. Appl. Anal. 2014, Article ID 601839 (2014) MathSciNetGoogle Scholar
  21. Zhou, L, Sun, J-J: Six new Redheffer-type inequalities for circular and hyperbolic functions. Comput. Math. Appl. 56(2), 522-529 (2008) MathSciNetView ArticleMATHGoogle Scholar
  22. Bhayo, BA, Sándor, J: On Jordan’s, Redheffer’s and Wilker’s inequality. http://files.ele-math.com/preprints/mia-4389-pre.pdf
  23. Mitrinović, DS, Pečarić, JE, Fink, AM: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht (1993) View ArticleMATHGoogle Scholar
  24. Qi, F, Guo, B-N: A class of logarithmically completely monotonic functions and the best bounds in the second Kershaw’s double inequality. J. Comput. Appl. Math. 212(2), 444-456 (2008) MathSciNetView ArticleMATHGoogle Scholar
  25. Qi, F: A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw’s double inequality. J. Comput. Appl. Math. 206(2), 1007-1014 (2007) MathSciNetView ArticleMATHGoogle Scholar

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