Abstract
We prove that the double inequalities \(H_{\alpha}(a, b) < X(a, b)<H_{\beta}(a, b)\) and \(H_{\lambda}(a, b)< U(a, b)<H_{\mu}(a, b)\) hold for all \(a, b>0\) with \(a\neq b\) if and only if \(\alpha\leq1/2\), \(\beta\geq\log3/(1+\log2)=0.6488\cdots\), \(\lambda\leq2\log3/(2\log\pi-\log2) =1.3764\cdots\), and \(\mu\geq2\), where \(H_{p}(a, b)\), \(X(a, b)\), and \(U(a, b)\) are, respectively, the pth power-type Heronian mean, Sándor mean, and Yang mean of a and b.
Similar content being viewed by others
1 Introduction
For \(p\in\mathbb{R}\), the pth power-type Heronian mean \(H_{p}(a, b)\) of two positive real numbers a and b is defined by
It is well known that \(H_{p}(a, b)\) is continuous and strictly increasing with respect to \(p\in\mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\).
Let \(G(a, b)=\sqrt{ab}\), \(L(a, b)=(a-b)/(\log a-\log b)\), \(P(a, b)=(a-b)/[2\arcsin((a-b)/(a+b))]\), \(I(a, b)=(a^{a}/b^{b})^{1/(a-b)}/e\), \(A(a, b)=(a+b)/2\), \(T(a, b)=(a-b)/[2\arctan((a-b)/(a+b))]\), \(Q(a, b)=\sqrt{(a^{2}+b^{2})/2}\) and \(M_{r}(a, b)=[(a^{r}+b^{r})/2]^{1/r}\) (\(r\neq0\)), and \(M_{0}(a, b)=\sqrt{ab}\) be, respectively, the geometric, logarithmic, first Seiffert, identric, arithmetic, second Seiffert, quadratic, and rth power means of two distinct positive real numbers a and b. Then it is well known that the inequalities
hold for all \(a, b>0\) with \(a\neq b\).
Let \(a, b>0\). Then the Sándor mean \(X(a,b)\) [1] and Yang mean \(U(a,b)\) [2] are given by
and
respectively.
The Yang mean \(U(a,b)\) is the special case of the Seiffert type mean \(T_{M, p}(a, b)=(a-b)/[p\arctan((a-b)/(pM(a,b)))]\) defined by Toader in [3], where \(M(a, b)\) is a bivariate mean and p is a positive real number. Indeed, \(U(a,b)=T_{G,\sqrt{2}}(a,b)\). Recently, the power-type Heronian, Sándor, and Yang means have been the subject of intensive research.
For all \(a, b>0\) with \(a\neq b\), Yang [4] and Sándor [5] proved that the double inequality
holds, and the inequality \(H_{1}(a,b)< M_{2/3}(a,b)\) can be found in the literature [6].
Jia and Cao [7] proved that the inequalities
hold for all \(a, b>0\) with \(a\neq b\) if \(p\geq1/2\) and \(q\geq2p/3\). Inequality (1.4) can also be found in the literature [8], p.64 and [9].
In [10], the authors proved that the double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq\log3/(\log \pi-\log2)\) and \(q\geq5/2\).
Sándor [11] presented the inequalities
for all \(a, b>0\) with \(a\neq b\).
Yang et al. [12] proved that the double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq1/3\) and \(q\geq\log2/(1+\log2)\).
In [2], Yang established the inequalities
for all \(a, b>0\) with \(a\neq b\).
In [13, 14], the authors proved that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq p_{0}\), \(q\geq1/5\), \(\lambda\geq1/5\), \(\mu\leq p_{1}\), \(\alpha\leq2\log 2/(2\log\pi-\log2)\), and \(\beta\geq4/3\), where \(p_{0}=0.1941\cdots\) is the unique solution of the equation \(p\log(2/\pi)-\log (1+2^{1-p} )+\log3=0\) on the interval \((1/10, \infty)\), and \(p_{1}=\log(\pi-2)/\log2=0.1910\cdots\).
The main purpose of this paper is to present the best possible parameter α, β, λ, and μ such that the double inequalities \(H_{\alpha}(a, b)< X(a, b)<H_{\beta}(a, b)\) and \(H_{\lambda}(a, b)< U(a, b)<H_{\mu}(a, b)\) 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 \(p\in(0, 1)\) and
Then the following statements are true:
-
(1)
if \(p=1/2\), then \(f(x)<0\) for all \(x>1\);
-
(2)
if \(p=\log3/(1+\log2)=0.6488\cdots\), then there exists \(\lambda \in(1, \infty)\) such that \(f(x)>0\) for \(x\in(1, \lambda)\) and \(f(x)<0\) for \(x\in(\lambda, \infty)\).
Proof
For part (1), if \(p=1/2\), then (2.1) becomes
Therefore, part (1) follows from (2.2).
For part (2), let \(p=\log3/(1+\log2)\), \(f_{1}(x)=f^{\prime}(x)/x\), \(f_{2}(x)=x^{5-p}f_{1}^{\prime}(x)\), \(f_{3}(x)=f_{2}^{\prime }(x)/(2px)\), and \(f_{4}(x)=f_{3}^{\prime}(x)/(2x)\). Then elaborated computations lead to
It follows from (2.7) and \(p=\log3/(1+\log2)=0.6488\cdots\) together with \(13p^{4}+337p^{3}-80p^{2}-72=-11.3153\cdots<0\) that
for \(x\in(1, \infty)\).
Inequality (2.8) implies that \(f_{3}(x)\) is strictly decreasing on \([1, \infty)\). Then (2.6) leads to the conclusion that there exists \(\lambda _{1}\in(1, \infty)\) such that \(f_{2}(x)\) is strictly increasing on \([1, \lambda_{1}]\) and strictly decreasing on \([\lambda_{1}, \infty)\).
It follows from the piecewise monotonicity of \(f_{2}\) and (2.5) that there exists \(\lambda_{2}\in(\lambda_{1}, \infty)\) such that \(f_{1}(x)\) is strictly increasing on \([1, \lambda_{2}]\) and strictly decreasing on \([\lambda_{2}, \infty)\).
From (2.4) and the piecewise monotonicity of \(f_{1}\) we clearly see that there exists \(\lambda_{3}\in(\lambda_{2}, \infty)\) such that \(f(x)\) is strictly increasing on \([1, \lambda_{3}]\) and strictly decreasing on \([\lambda_{3}, \infty)\).
Therefore, part (2) follows from (2.3) and the piecewise monotonicity of f. □
Lemma 2.2
Let \(p\in(1, 2]\) and
Then the following statements are true:
-
(1)
if \(p=2\), then \(g(x)>0\) for all \(x>1\);
-
(2)
if \(p=2\log3/(2\log\pi-\log2)=1.3764\cdots\), then there exists \(\mu\in(1, \infty)\) such that \(g(x)<0\) for \(x\in(1, \mu)\) and \(g(x)>0\) for \(x\in(\mu, \infty)\).
Proof
For part (1), if \(p=2\), then (2.9) becomes
Therefore, part (1) follows from (2.10).
For part (2), let \(p=2\log3/(2\log\pi-\log2)\), \(g_{1}(x)=g^{\prime }(x)/x\), \(g_{2}(x)=g_{1}^{\prime}(x)/x\), \(g_{3}(x)=g_{2}^{\prime }(x)/x\), and \(g_{4}(x)=x^{9-p}g_{3}^{\prime}(x)\). Then elaborated computations lead to
Note that
It follows from (2.15)-(2.20) that
for \(x\in(1, \infty)\).
Therefore, part (2) follows easily from (2.11)-(2.14) and (2.21). □
3 Main results
Theorem 3.1
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(\alpha\leq1/2\) and \(\beta\geq\log3/(1+\log2)=0.6488\cdots\).
Proof
Since \(X(a,b)\) and \(H_{p}(a,b)\) are symmetric and homogeneous of degree one, we assume that \(a>b\). Let \(x=\sqrt{a/b}\in (1, \infty)\) and \(p\in\mathbb{R}\). Then (1.1) and (1.2) lead to
Simple computations lead to
where
where \(f(x)\) is defined as in Lemma 2.1.
If \(p=1/2\), then from Lemma 2.1(1), (3.1), (3.2), and (3.4)-(3.6) we clearly see that
for all \(a, b>0\) with \(a\neq b\).
If \(p=\log3/(1+\log2)\), then (3.3) becomes
It follows from Lemma 2.2(2) and (3.6) that there exists \(\lambda\in (1, \infty)\) such that \(F_{1}(x)\) is strictly decreasing on \([1, \lambda ]\) and strictly increasing on \([\lambda, \infty)\).
Equations (3.4) and (3.5) together with the piecewise monotonicity of \(F_{1}\) lead to the conclusion that there exists \(\lambda^{\ast}\in(1, \infty)\) such that \(F(x)\) is strictly decreasing on \([1, \lambda^{\ast }]\) and strictly increasing on \([\lambda^{\ast}, \infty)\).
Therefore,
for all \(a, b>0\) with \(a\neq b\) follows from (3.1), (3.2), (3.7), and the piecewise monotonicity of F.
Next, we prove that \(\alpha=1/2\) and \(\beta=\log3/(1+\log2)\) are the best possible parameters such that the double inequality \(H_{\alpha}(a, b)< X(a, b)<H_{\beta}(a, b)\) holds for all \(a, b>0\) with \(a\neq b\).
If \(p<\log3/(1+\log2)\), then (3.3) leads to
Equation (3.1) and inequality (3.8) imply that there exists large enough \(T_{0}=T_{0}(p)>1\) such that \(X(a,b)>H_{p}(a,b)\) for all \(a, b>0\) with \(a/b\in(T_{0}, \infty)\) if \(p<\log3/(1+\log2)\).
Let \(p>1/2\), \(x>0\), and \(x\rightarrow0\). Then elaborated computations lead to
Equation (3.9) implies that there exists small enough \(\delta_{0}=\delta _{0}(p)>0\) such that \(X(1, 1+x)< H{_{p}}(1, 1+x)\) for \(x\in(0, \delta _{0})\) if \(p>1/2\). □
Theorem 3.2
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda\leq2\log 3/(2\log\pi-\log2)=1.3764\cdots\) and \(\mu\geq2\).
Proof
Since \(U(a,b)\) and \(H_{p}(a,b)\) are symmetric and homogeneous of degree one, we assume that \(a>b\). Let \(x=\sqrt{a/b}\in (1, \infty)\) and \(p\in\mathbb{R}\). Then (1.1) and (1.3) lead to
Simple computations lead to
where
where \(g(x)\) is defined as in Lemma 2.2.
If \(p=2\log3/(2\log\pi-\log2)\), then (3.15) and Lemma 2.2(2) lead to the conclusion that there exists \(\mu\in(1, \infty)\) such that \(G_{1}(x)\) is strictly increasing on \([1, \mu]\) and strictly decreasing on \([\mu, \infty)\).
It follows from (3.13) and (3.14) together with the piecewise monotonicity of \(G_{1}\) that there exists \(\mu^{\ast}\in(1, \infty)\) such that \(G(x)\) is strictly increasing on \([1, \mu^{\ast}]\) and strictly decreasing on \([\mu^{\ast}, \infty)\).
Note that (3.12) becomes
Therefore,
for all \(a, b>0\) with \(a\neq b\) follows from (3.10), (3.11), and (3.16) together with the piecewise monotonicity of G.
If \(p=2\), then
for all \(a, b>0\) with \(a\neq b\) follows easily from (3.10), (3.11), and (3.13)-(3.15) together with Lemma 2.2(1).
Next, we prove that \(\lambda=2\log3/(2\log\pi-\log2)\) and \(\mu=2\) are the best possible parameters such that the double inequality
holds for all \(a, b>0\) with \(a\neq b\).
If \(p>2\log3/(2\log\pi-\log2)\), then (3.12) leads to
Equation (3.10) and inequality (3.17) imply that there exists large enough \(T_{1}=T_{1}(p)>1\) such that \(U(a, b)< H_{p}(a,b)\) for all \(a, b>0\) with \(a/b\in(T_{1}, \infty)\).
Let \(p<2\), \(x>0\), and \(x\rightarrow0\). Then elaborated computations lead to
Inequality (3.18) implies that there exists small enough \(\delta _{1}=\delta_{1}(p)>0\) such that \(U(1, 1+x)>H_{p}(1, 1+x)\) for \(x\in(0, \delta_{1})\). □
References
Sándor, J: Two sharp inequalities for trigonometric and hyperbolic functions. Math. Inequal. Appl. 15(2), 409-413 (2012)
Yang, Z-H: Three families of two-parameter means constructed by trigonometric functions. J. Inequal. Appl. 2013, Article ID 541 (2013)
Toader, G: Seiffert type means. Nieuw Arch. Wiskd. 17(3), 379-382 (1999)
Yang, Z-H: The exponential mean and the logarithmic mean. Math. Pract. Theory 4, 76-78 (1987) (in Chinese)
Sándor, J: A note on some inequalities for means. Arch. Math. 56(5), 471-473 (1991)
Kuang, J-C: Applied Inequalities. Hunan Education Press, Changsha (1993) (in Chinese)
Jia, G, Cao, J-D: A new upper bound of the logarithmic mean. JIPAM. J. Inequal. Pure Appl. Math. 4(4), 80 (2003)
Kuang, J-C: Applied Inequalities. Shandong Science Technology Press, Jinan (2010) (in Chinese)
Alzer, H, Janous, W: Solution of problem 8∗. Crux Math. 13, 173-178 (1987)
Chu, Y-M, Wang, M-K, Qiu, Y-F: An optimal double inequality between power-type Heron and Seiffert means. J. Inequal. Appl. 2010, Article ID 146945 (2010)
Sándor, J: On two new means of two variables. Notes Number Theory Discrete Math. 20(1), 1-9 (2014)
Yang, Z-H, Wu, L-M, Chu, Y-M: Sharp power mean bounds for Sándor mean. Abstr. Appl. Anal. 2014, Article ID 172867 (2014)
Yang, Z-H, Chu, Y-M, Song, Y-Q, Li, Y-M: A sharp double inequality for trigonometric functions and its applications. Abstr. Appl. Anal. 2014, Article ID 592085 (2014)
Yang, Z-H, Wu, L-M, Chu, Y-M: Sharp power mean bounds for Yang mean. J. Inequal. Appl. 2014, Article ID 401 (2014)
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 61374086, 11171307 and 11401191, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, the Natural Science Foundation of the Open University of China under Grant Q1601E-Y and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-13Z04.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Zhou, SS., Qian, WM., Chu, YM. et al. Sharp power-type Heronian mean bounds for the Sándor and Yang means. J Inequal Appl 2015, 159 (2015). https://doi.org/10.1186/s13660-015-0683-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0683-7