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.
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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
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 [2–13]. 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:
where
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
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
Then elaborated computations lead to
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
and
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
and (2.1) leads
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
□
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
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
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
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
listed in [18], 4.3.71, we get
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
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
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
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
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
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
Then we clearly see that
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
Therefore,
for all \(u>0\) if and only if \(r\leq\min\{s, t\}\), and
for all \(u>0\) if and only if \(r\geq(s+t)/2\).
From (3.1) and the integral formulas
given in [18], 6.1.50, 6.3.21, we get
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
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
and
Then we clearly see that
and
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
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
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
we have
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
for all \(x>0\).
Proof
It follows from the third inequality in Corollary 2.2 that
for \(u>0\).
Let
Then
It follows from (3.13)-(3.15) that
for \(x>0\).
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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.
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Yang, ZH., Chu, YM. Monotonicity and absolute monotonicity for the two-parameter hyperbolic and trigonometric functions with applications. J Inequal Appl 2016, 200 (2016). https://doi.org/10.1186/s13660-016-1143-8
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DOI: https://doi.org/10.1186/s13660-016-1143-8