Abstract
In this paper, we present the best possible parameters p and q such that the double inequality
holds for all \(a, b>0\) with \(a\neq b\), where \(M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}\) (\(r\neq0\)) and \(M_{0}(a,b)= \sqrt {ab}\) is the rth power mean and \(V(a,b)=(a-b)/[\sqrt{2}\sinh^{-1}((a-b)/\sqrt{2ab})]\) is the second Yang mean.
Similar content being viewed by others
1 Introduction
For \(r\in\mathbb{R}\), the rth power mean \(M_{r}(a,b)\) of two distinct positive real numbers a and b is defined by
It is well known that \(M_{r}(a,b)\) is continuous and strictly increasing with respect to \(r\in\mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\). Many classical means are special cases of the power mean, for example, \(M_{-1}(a,b)=2ab/(a+b)=H(a,b)\) is the harmonic mean, \(M_{0}(a,b)=\sqrt{ab}=G(a,b)\) is the geometric mean, \(M_{1}(a,b)=(a+b)/2=A(a,b)\) is the arithmetic mean, and \(M_{2}(a,b)=\sqrt{(a^{2}+b^{2})/2}=Q(a,b)\) is the quadratic mean. The main properties for the power mean are given in [1].
Let
and
be, respectively, the logarithmic mean, identric mean, first Seiffert mean [2], first Yang mean [3], Toader mean [4], Neuman-Sándor mean [5, 6], Sándor mean [7], second Seiffert mean [8], Sándor-Yang mean [3], and second Yang mean [3] of two distinct positive real numbers a and b, where \(\sinh^{-1}(x)=\log(x+\sqrt {x^{2}+1})\) is the inverse hyperbolic sine function.
Recently, the bounds for certain bivariate means in terms of the power mean have attracted the attention of many mathematicians. Radó [9] (see also [10–12]) proved that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq0\), \(q\geq 1/3\), \(\lambda\leq2/3\), and \(\mu\geq\log2\).
In [13–16], the authors proved that the double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq3/2\) and \(q\geq\log2/(\log\pi-\log2)\).
Jagers [17], Hästö [18, 19], Yang [20], and Costin and Toader [21] proved that \(p_{1}=\log2/\log\pi\), \(q_{1}=2/3\), \(p_{2}=\log 2/(\log\pi-\log2)\), and \(q_{2}=5/3\) are the best possible parameters such that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\).
In [21–26], the authors proved that the double inequalities
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 \(\mu_{3}\geq\log2/(1+\log2)\).
Very recently, Yang and Chu [27] showed that \(p=4\log2/(4+2\log2-\pi )\) and \(q=4/3\) are the best possible parameters such that the double inequality
holds for all \(a, b>0\) with \(a\neq b\).
The main purpose of this paper is to present the best possible parameters p and q such that the double inequality
holds for all \(a, b>0\) with \(a\neq b\).
2 Lemmas
In order to prove our main results we need three lemmas, which we present in this section.
Lemma 2.1
Let \(t>0\), \(p\in\mathbb{R}\), and
Then the following statements are true:
-
(i)
\(f(t,p)>0\) for all \(t>0\) if and only if \(p\geq2/3\);
-
(ii)
\(f(t,p)<0\) for all \(t>0\) if and only if \(p\leq0\).
Proof
It follows from (2.1) that
for all \(t>0\) and \(p\in\mathbb{R}\).
(i) If \(f(t,p)>0\) for all \(t>0\), then (2.1) leads to
which gives \(p\geq2/3\).
If \(p\geq2/3\), then (2.1) and (2.2) lead to the conclusion that
for all \(t>0\).
(ii) If \(f(t, p)<0\) for all \(t>0\), then from part (i) we know that \(p<2/3\). We assert that \(p\leq0\), otherwise \(0< p<2/3\) and (2.1) leads to
which contradicts with \(f(t, p)<0\) for all \(t>0\).
If \(p\leq0\), then from (2.1) and (2.2) we have
for all \(t>0\). □
Lemma 2.2
The double inequality
holds for all \(t>0\) if and only if \(p\leq0\) and \(q\geq2/3\). Here
Proof
Let \(t>0\), \(p\in\mathbb{R}\) and \(F(t, p)\) be defined by
Then making use of the power series formulas
we get
for \(t\rightarrow0^{+}\).
It follows from (2.4) and (2.5) that
where
where \(f(t,p)\) is defined by Lemma 2.1.
and
if \(p>0\).
We first prove that the inequality
holds for all \(t>0\) if and only if \(p\geq2/3\).
If \(p\geq2/3\), then inequality (2.13) holds for all \(t>0\) follows easily from Lemma 2.1(i), (2.4), (2.6), (2.7), (2.9), and (2.10).
If inequality (2.13) holds for all \(t>0\), then (2.4) and (2.11) lead to \(p\geq2/3\).
Next, we prove that the inequality
holds for all \(t>0\) if and only if \(p\leq0\).
If \(p\leq0\), then that inequality (2.14) holds for all \(t>0\) follows easily from Lemma 2.1(ii), (2.4), (2.6), (2.7), (2.9), and (2.10).
If inequality (2.14) holds for all \(t>0\), then (2.4) leads to \(F(t,p)>0\). We assert that \(p\leq0\), otherwise \(p>0\) and (2.12) implies that there exists large enough \(T_{0}>0\) such that \(F(t, p)<0\) for \(t\in(T_{0}, \infty)\). □
Lemma 2.3
Let \(t>0\), \(p\in\mathbb{R}\), and \(f_{1}(t,p)\) be defined by (2.8). Then the following statements are true:
-
(i)
\(f_{1}(t,p)<0\) for all \(t>0\) if and only if \(p\geq2/3\);
-
(ii)
\(f(t,p)>0\) for all \(t>0\) if and only if \(p\leq0\).
Proof
(i) If \(p\geq2/3\), then \(f_{1}(t,p)<0\) for all \(t>0\) follows easily from (2.9) and (2.10) together with Lemma 2.1(i).
If \(f_{1}(t,p)<0\) for all \(t>0\), then (2.8) leads to
which gives \(p\geq2/3\).
(ii) If \(p\leq0\), then \(f_{1}(t,p)>0\) for all \(t>0\) follows easily from (2.9) and (2.10) together with Lemma 2.1(ii).
Note that
If \(f_{1}(t,p)>0\) for all \(t>0\), then
and we assert that \(p\leq0\). Otherwise, equation (2.15) leads to
if \(p=1\) and
if \(p\in(0, 1)\cup(1, \infty)\). □
3 Main results
Theorem 3.1
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq0\) and \(q\geq2/3\).
Proof
Since both \(M_{r}(a,b)\) and \(V(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(a>b>0\). Let \(t=\frac{1}{2}\log(a/b)>0\) and \(r\in\mathbb{R}\), then (1.1) and (1.2) lead to
and
Therefore, Theorem 3.1 follows easily from (3.1) and (3.2) together with Lemma 2.2. □
Theorem 3.2
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\geq2/3\) and \(q\leq0\).
Proof
Without loss of generality, we assume that \(a>b>0\). Let \(t=\frac{1}{2}\log(a/b)>0\) and \(r\in\mathbb{R}\), then
Therefore, Theorem 3.2 follows easily from (3.1) and (3.3) together with Lemma 2.3. □
Let \(p\in\mathbb{R}\) and \(a, b>0\). Then the pth Lehmer mean [28] \(L_{p}(a,b)=\frac{a^{p+1}+b^{p+1}}{a^{p}+b^{p}}\) is strictly increasing with respect to \(p\in\mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\). From Theorem 3.2 we get Corollary 3.3 immediately.
Corollary 3.3
The double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\geq2/3\) and \(q\leq0\).
Let \(p=2/3, 1, 2, +\infty\) and \(q=0, -1/2, -1, -2, -\infty\). Then Corollary 3.3 leads to
Corollary 3.4
The inequalities
hold for all \(a, b>0\) with \(a\neq b\).
From (1.3), (1.4), and Theorem 3.1 we clearly see that \(M_{2/3}(a,b)\) is the sharp upper power mean bound for the 2-order generalized logarithmic mean \(L^{1/2}(a^{2}, b^{2})\), the first Seiffert mean \(P(a,b)\), and the second Yang mean \(V(a,b)\). In [29], Theorem 3, Yang and Chu proved that the inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(r\leq2\).
As a result of comparing \(V(a,b)\) with \(L^{1/2} (a^{2}, b^{2} )\), we have the following.
Theorem 3.5
The inequality
holds for all \(a, b>0\) with \(a\neq b\).
Proof
We assume that \(a>b\). Let \(t=\frac{1}{2}\log(a/b)>0\), then
It follows from (3.1) and (3.5) that
Let
Then simple computation leads to
for \(t>0\).
Therefore, Theorem 3.5 follows easily from (3.6)-(3.10). □
Remark 3.6
From (1.4), (3.4), Theorems 3.1, and 3.5 we get the inequalities
for all \(a, b>0\) with \(a\neq b\).
References
Bullen, PS, Mitrinović, DS, Vasić, PM: Means and Their Inequalities. Reidel, Dordrecht (1988)
Seiffert, H-J: Problem 887. Nieuw Arch. Wiskd. 11(2), 176 (1993)
Yang, Z-H: Three families of two-parameter means constructed by trigonometric functions. J. Inequal. Appl. 2013, Article ID 541 (2013)
Toader, G: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218(2), 358-368 (1998)
Neuman, E, Sándor, J: On the Schwab-Borchardt mean. Math. Pannon. 14(2), 253-266 (2003)
Neuman, E, Sándor, J: On the Schwab-Borchardt mean II. Math. Pannon. 17(1), 49-59 (2006)
Sándor, J: Two sharp inequalities for trigonometric and hyperbolic functions. Math. Inequal. Appl. 15(2), 409-413 (2012)
Seiffert, H-J: Aufgabe β16. Die Wurzel 29, 221-222 (1995)
Radó, T: On convex functions. Trans. Am. Math. Soc. 37(2), 266-285 (1935)
Lin, TP: The power mean and the logarithmic mean. Am. Math. Mon. 81, 879-883 (1974)
Stolarsky, KB: The power and generalized logarithmic means. Am. Math. Mon. 87(7), 545-548 (1980)
Pittenger, AO: Inequalities between arithmetic and logarithmic means. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz. 678-715, 15-18 (1980)
Qiu, S-L, Shen, J-M: On two problems concerning means. J. Hangzhou Inst. Electron. Eng. 17(3), 1-7 (1997) (in Chinese)
Qiu, S-L: The Muir mean and the complete elliptic integral of the second kind. J. Hangzhou Inst. Electron. Eng. 20(1), 28-33 (2000) (in Chinese)
Barnard, RW, Pearce, K, Richards, KC: An inequality involving the generalized hypergeometric function and the arc length of an ellipse. SIAM J. Math. Anal. 31(3), 693-699 (2000)
Alzer, H, Qiu, S-L: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172(2), 289-312 (2004)
Jagers, AA: Solution of problem 887. Nieuw Arch. Wiskd. 12(2), 230-231 (1994)
Hästö, PA: A monotonicity property of ratios of symmetric homogeneous means. JIPAM. J. Inequal. Pure Appl. Math. 3(5), Article ID 71 (2002)
Hästö, PA: Optimal inequalities between Seiffert’s mean and power means. Math. Inequal. Appl. 7(1), 47-53 (2004)
Yang, Z-H: Sharp bounds for the second Seiffert mean in terms of power means. arXiv:1206.5494 [math.CA]
Costin, I, Toader, G: Optimal evaluations of some Seiffert-type means by power means. Appl. Math. Comput. 219(9), 4745-4754 (2013)
Yang, Z-H: Sharp power means bounds for Neuman-Sándor mean. arXiv:1208.0895 [math.CA]
Yang, Z-H: Estimates for Neuman-Sándor mean by power means and their relative errors. J. Math. Inequal. 7(4), 711-726 (2013)
Chu, Y-M, Long, B-Y: Bounds of the Neuman-Sándor mean using power and identric means. Abstr. Appl. Anal. 2013, Article ID 832591 (2013)
Yang, Z-H, Wu, L-M, Chu, Y-M: Optimal power mean bounds for Yang mean. J. Inequal. Appl. 2014, Article ID 401 (2014)
Chu, Y-M, Yang, Z-H, Wu, L-M: Sharp power mean bounds for Sándor mean. Abstr. Appl. Anal. 2015, Article ID 172867 (2015)
Yang, Z-H, Chu, Y-M: Optimal evaluations for the Sándor-Yang mean by power mean. arXiv:1506.07777 [math.CA]
Lehmer, DH: On the compounding of certain means. J. Math. Anal. Appl. 36(4), 183-200 (1971)
Yang, Z-H, Chu, Y-M: An optimal inequalities chain for bivariate means. J. Math. Inequal. 9(2), 331-343 (2015)
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 Major Project Foundation of the Department of Education of Hunan Province under Grant 12A026.
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
Li, JF., Yang, ZH. & Chu, YM. Optimal power mean bounds for the second Yang mean. J Inequal Appl 2016, 31 (2016). https://doi.org/10.1186/s13660-016-0970-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-016-0970-y