Abstract
In this paper, we find the greatest values α, λ and the least values β, μ such that the double inequalities \(\alpha[G(a,b)/3+2A(a,b)/3]+(1-\alpha)G^{1/3}(a,b)A^{2/3}(a,b)< P(a,b) <\beta[G(a,b)/3+2A(a,b)/3]+(1-\beta)G^{1/3}(a,b)A^{2/3}(a,b)\) and \(\lambda [C(a,b)/3+2A(a,b)/3 ]+ (1-\lambda)C^{1/3}(a,b) A^{2/3}(a,b)< T(a,b)<\mu [C(a,b)/3+2A(a,b)/3 ]+(1-\mu)C^{1/3}(a,b) A^{2/3}(a,b)\) hold for all \(a,b>0\) with \(a\neq b\). Here \(G(a,b)\), \(A(a,b)\), \(C(a,b)\), \(P(a,b)\) and \(T(a,b)\) denote the geometric, arithmetic, contraharmonic, first Seiffert and second Seiffert means of two positive numbers a and b, respectively.
Similar content being viewed by others
1 Introduction
For \(a,b>0\) with \(a\neq b\), the first and second Seiffert means \(P(a,b)\) [1] and \(T(a,b)\) [2] are defined by
and
respectively.
Recently, both means P and T have been the subject of intensive research. In particular, many remarkable inequalities for P and T can be found in the literature [3–9]. The first Seiffert mean \(P(a,b)\) can be rewritten as (see [10, Eq. (2.4)])
Let \(H(a,b)={2ab}/({a+b})\), \(G(a,b)=\sqrt{ab}\), \(L(a,b)=(b-a)/(\log b-\log a)\), \(I(a,b)={1}/{e}({b^{b}}/{a^{a}})^{{1}/({b-a})}\), \(A(a,b)=(a+b)/2\), \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\), \(C(a,b)=(a^{2}+b^{2})/(a+b)\), \(L_{r}(a,b)=(a^{r+1}+b^{r+1})/(a^{r}+b^{r})\), and \(M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}\) (\(r\neq0\)) and \(M_{0}(a,b)=G(a,b)\) be the harmonic, geometric, logarithmic, identric, arithmetic, quadratic, contraharmonic, rth Lehmer and rth power means of two distinct positive real numbers a and b, respectively. Then both \(L_{r}(a,b)\) and \(M_{r}(a,b)\) are strictly increasing with respect to \(r\in\mathbb{R}\) for fixed \(a, b>0\) with \(a\neq b\), and the inequalities
hold for all \(a, b>0\) with \(a\neq b\).
Jagers [11] and Seiffert [2] proved that the inequalities
hold for \(a,b>0\) with \(a\neq b\).
Costin and Toader [12] proved that the double inequality
holds for \(a,b>0\) with \(a\neq b\).
In [13–17], the authors proved that the inequalities
hold for \(a,b>0\) with \(a\neq b\) if and only if \(p\leq\log\pi/\log 2\), \(q\geq2/3\), \(r\leq\log2/(\log\pi-\log2)\), \(s\geq5/3\), \(\alpha\leq-1/6\), \(\beta\geq0\), \(\sigma\leq0\), \(\tau\geq1/3\), \(\lambda\geq2\) and \(\mu\geq5\).
Gao [18] proved that \(\alpha=e/\pi\), \(\beta=1\), \(\lambda=1\) and \(\mu=2e/\pi\) are the best possible constants such that the double inequalities
hold for \(a,b>0\) with \(a\neq b\).
In [19, 20], the authors proved that the double inequalities
hold for \(a,b>0\) with \(a\neq b\) if and only if \(\alpha_{1}\leq2/9\), \(\beta_{1}\geq1/\pi\), \(\alpha_{2}\leq2/\pi\), \(\beta_{2}\geq2/3\), \(\alpha_{3}\leq2/\pi\) and \(\beta_{3}\geq2/3\).
Let \(p\geq1/2\), \(q\geq1\), \(t_{1}, t_{2}\in(1/2, 1)\) and \(t_{3}, t_{4}\in(0, 1/2)\). Then the authors in [21, 22] proved that the double inequalities
hold for \(a,b>0\) with \(a\neq b\) if and only if \(t_{1}\leq [1+\sqrt{(4/\pi)^{1/p}-1}]/2\), \(t_{2}\geq1/2+\sqrt{3p}/(6p)\), \(t_{3}\leq[1-\sqrt{1-(2/\pi)^{2/q}}]/2\) and \(t_{4}\geq (1-1/\sqrt{3q})/2\).
Yang et al. [23] proved that the double inequality
holds for \(a,b>0\) with \(a\neq b\) if and only if \(p\geq5/3\) and \(q\leq1\).
Sándor [24] and Jiang et al. [25] proved that the inequalities
hold for \(a,b>0\) with \(a\neq b\).
In [26], Sándor found that \(T(a,b)\) is the common limit of the sequences \(\{u_{n}\}\) and \(\{v_{n}\}\) given by
and established a more general inequality
for all \(n\geq0\) and \(a, b>0\) with \(a\neq b\). In particular, let \(n=0\), then (1.4) and (1.5) together with the identity \(Q^{2/3}(a,b)A^{1/3}(a,b)=C^{1/3}(a,b)A^{2/3}(a,b)\) lead to
for all \(a, b>0\) with \(a\neq b\).
Motivated by inequalities (1.3) and (1.6), we naturally ask: what are the best possible parameters α, β, λ and μ such that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\)? The purpose of this paper is to answer this question.
2 Lemmas
In order to establish our main results, we need two lemmas, which we present in this section.
Lemma 2.1
Let \(g(t)=-p^{2}t^{6}-2p^{2}t^{5}+3(p^{2}-4p+2)t^{4}+2(2p^{2}-9p+6)t^{3}-(4p^{2}+6p-9)t^{2}+6(1-p)t+3(1-p)\). Then the following statements are true:
-
(1)
If \(p=4/5\), then \(g(t)>0\) for all \(t\in(0,1)\).
-
(2)
If \(p=3/\pi\), then there exists \(\lambda_{0}\in(0,1)\) such that \(g(t)>0\) for \(t\in(0,\lambda_{0})\) and \(g(t)<0\) for \(t\in(\lambda_{0},1)\).
Proof
Part (1) follows easily from
for all \(t\in(0,1)\) if \(p=4/5\).
For part (2), if \(p=3/\pi\), then numerical computations lead to
It follows from (2.1)-(2.3) and (2.8) that \(g^{\prime}\) is strictly decreasing on \((0, 1)\). Then (2.6) and (2.7) lead to the conclusion that there exists \(\lambda_{1}\in(0, 1)\) such that g is strictly increasing on \((0, \lambda_{1}]\) and strictly decreasing on \([\lambda_{1}, 1)\).
Therefore, part (2) follows from (2.4) and (2.5) together with the piecewise monotonicity of \(g^{\prime}\). □
Lemma 2.2
Let \(h(t)=q(q+3)t^{4}+2q(q+3)t^{3}-3(q^{2}-6q+1)t^{2}-2(2q^{2}-9q+3)t+4q^{2}\). Then the following statements are true:
-
(1)
If \(q=1/5\), then \(h(t)>0\) for \(t\in(1,\sqrt[3]{2})\).
-
(2)
If \(q=[3(\sqrt[3]{2}\pi-4)]/[(3\sqrt[3]{2}-4)\pi]=0.1814\ldots\) , then there exists \(\mu_{0}\in(1, \sqrt[3]{2})\) such that \(h(t)<0\) for \(t\in(1, \mu_{0})\) and \(h(t)>0\) for \(t\in(\mu_{0}, \sqrt[3]{2})\).
Proof
Part (1) follows easily from
for all \(t\in(1,\sqrt[3]{2})\) if \(q=1/5\).
For part (2), if \(q=[3(\sqrt[3]{2}\pi-4)]/[(3\sqrt[3]{2}-4)\pi]\), then numerical computations lead to
It follows from (2.9) and (2.12) that
for \(t\in(1,\sqrt[3]{2})\).
Therefore, part (2) follows easily from (2.10) and (2.11) together with (2.13). □
3 Main results
Theorem 3.1
The double inequality
holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\alpha\leq 4/5\) and \(\beta\geq3/\pi\).
Proof
Firstly, we prove that the inequalities
hold for all \(a,b>0\) with \(a\neq b\).
Since \(P(a,b)\), \(A(a,b)\) and \(G(a,b)\) are symmetric and homogenous of degree 1, without loss of generality, we assume that \(a>b\). Let \(x=(a-b)/(a+b)\in(0, 1)\) and \(p\in(0, 1)\). Then (1.2) leads to
where
where the function \(g(\cdot)\) is defined as in Lemma 2.1.
We divide the proof into two cases.
Case 1. \(p=4/5\). Then (3.1) follows easily from (3.6), (3.7), (3.9) and Lemma 2.1(1).
Case 2. \(p=3/\pi\). Then Lemma 2.1(2) and (3.9) lead to the conclusion that there exists \(x_{0}\in(0, 1)\) such that G is strictly decreasing on \((0, x_{0}]\) and strictly increasing on \([x_{0}, 1)\).
Note that (3.8) becomes
It follows from (3.7) and (3.10) together with the piecewise monotonicity of G that
for all \(x\in(0, 1)\).
Therefore, (3.2) follows from (3.6) and (3.11), and Theorem 3.1 follows from (3.1) and (3.2) in conjunction with the following statements.
-
If \(\alpha>4/5\), then equations (3.3) and (3.4) lead to the conclusion that there exists \(0<\delta_{1}<1\) such that \(P(a,b)<\alpha [G(a,b)/3+2A(a,b)/3 ]+(1-\alpha )G^{1/3}(a,b)A^{2/3}(a,b)\) for all \(a,b>0\) with \((a-b)/(a+b)\in(0,\delta_{1})\).
-
If \(\beta<3/\pi\), then equations (3.3) and (3.5) imply that there exists \(0<\delta_{2}<1\) such that \(P(a,b)>\beta [G(a,b)/3+2A(a,b)/3 ]+(1-\beta)G^{1/3}(a,b)A^{2/3}(a,b)\) for all \(a,b>0\) with \((a-b)/(a+b)\in(1-\delta_{2},1)\).
□
Theorem 3.2
The double inequality
holds for all \(a,b>0\) with \(a\neq b\) if and only if \(\lambda\leq [3(\sqrt[3]{2}\pi-4)]/[(3\sqrt[3]{2}-4)\pi]=0.1814\ldots\) and \(\mu \geq1/5\).
Proof
Let \(\lambda^{\ast}=[3(\sqrt[3]{2}\pi-4)]/[(3\sqrt [3]{2}-4)\pi]\). Firstly, we prove that the inequalities
hold for all \(a,b>0\) with \(a\neq b\).
Since \(T(a,b)\), \(A(a,b)\) and \(C(a,b)\) are symmetric and homogenous of degree 1, without loss of generality, we assume that \(a>b\). Let \(x=(a-b)/(a+b)\in(0, 1)\) and \(q\in(0, 1)\). Then (1.1) leads to
where
where the function \(h(\cdot)\) is defined as in Lemma 2.2.
We divide the proof into two cases.
Case 1. \(q=1/5\). Then (3.12) follows easily from Lemma 2.2(1), (3.17), (3.18) and (3.20).
Case 2. \(q=\lambda^{\ast}\). Then Lemma 2.2(2) and (3.20) lead to the conclusion that there exists \(x^{\ast}\in(0, 1)\) such that H is strictly increasing on \((0, x^{\ast}]\) and strictly decreasing on \([x^{\ast}, 1)\).
Note that (3.19) becomes
Therefore, (3.13) follows from (3.17), (3.18), (3.21) and the piecewise monotonicity of H, and Theorem 3.2 follows from (3.12) and (3.13) in conjunction with the following statements.
-
If \(\mu<1/5\), then equations (3.14) and (3.15) lead to the conclusion that there exists \(0<\delta_{3}<1\) such that \(T(a,b)>\mu [C(a,b)/3+2A(a,b)/3 ]+(1-\mu)C^{1/3}(a,b)A^{2/3}(a,b)\) for all \(a,b>0\) with \((a-b)/(a+b)\in(0,\delta_{3})\).
-
If \(\lambda>\lambda^{\ast}\), then equations (3.14) and (3.16) imply that there exists \(0<\delta_{4}<1\) such that \(T(a,b)<\lambda [C(a,b)/3+2A(a,b)/3 ]+(1-\lambda )C^{1/3}(a,b)A^{2/3}(a,b)\) for all \(a,b>0\) with \((a-b)/(a+b)\in(1-\delta_{4},1)\).
□
References
Seiffert, H-J: Problem 887. Nieuw Arch. Wiskd. 11(2), 176 (1993)
Seiffert, H-J: Aufgabe β16. Wurzel 29, 221-222 (1995)
Chu, Y-M, Hou, S-W: Sharp bounds for Seiffert mean in terms of contraharmonic mean. Abstr. Appl. Anal. 2012, Article ID 425175 (2012)
Chu, Y-M, Wang, M-K, Wang, Z-K: Best possible inequalities among harmonic, geometric, logarithmic and Seiffert means. Math. Inequal. Appl. 15(2), 415-422 (2012)
Costin, I, Toader, G: A nice separation of some Seiffert-type means by power means. Int. J. Math. Math. Sci. 2012, Article ID 430692 (2012)
Jiang, W-D: Some sharp inequalities involving reciprocals of the Seiffert and other means. J. Math. Inequal. 6(4), 593-599 (2012)
Witkowski, A: Optimal weighted harmonic interpolations between Seiffert means. Colloq. Math. 130(2), 265-279 (2013)
Matejíčka, L: Sharp bounds for the weighted geometric mean of the first Seiffert and logarithmic means in terms of weighted generalized Heronian mean. Abstr. Appl. Anal. 2013, Article ID 721539 (2013)
Yang, Z-H: Sharp bounds for Seiffert mean in terms of weighted power means of arithmetic mean and geometric mean. Math. Inequal. Appl. 17(2), 499-511 (2014)
Neuman, E, Sándor, J: On the Schwab-Borchardt mean. Math. Pannon. 14(2), 253-266 (2003)
Jagers, AA: Solution of Problem 887. Nieuw Arch. Wiskd. 12, 230-231 (1994)
Costin, I, Toader, G: A separation of some Seiffert-type means by power means. Rev. Anal. Numér. Théor. Approx. 41(2), 125-129 (2012)
Hästö, PA: Optimal inequalities between Seiffert’s mean and power means. Math. Inequal. Appl. 7(1), 47-53 (2004)
Costin, I, Toader, G: Optimal evaluations of some Seiffert-type means by power means. Appl. Math. Comput. 219(9), 4745-4754 (2013)
Li, Y-M, Wang, M-K, Chu, Y-M: Sharp power mean bounds for Seiffert mean. Appl. Math. J. Chin. Univ. Ser. B 29(1), 101-107 (2014)
Wang, M-K, Qiu, Y-F, Chu, Y-M: Sharp bounds for Seiffert means in terms of Lehmer means. J. Math. Inequal. 4(4), 581-586 (2010)
Chu, Y-M, Long, B-Y, Gong, W-M, Song, Y-Q: Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means. J. Inequal. Appl. 2013, Article ID 10 (2013)
Gao, S-Q: Inequalities for the Seiffert’s means in terms of the identric mean. J. Math. Sci. Adv. Appl. 10(1-2), 23-31 (2011)
Liu, H, Meng, X-J: The optimal convex combination bounds for Seiffert’s mean. J. Inequal. Appl. 2011, Article ID 686834 (2011)
Jiang, W-D, Qi, F: Some sharp inequalities involving Seiffert and other means and their concise proofs. Math. Inequal. Appl. 15(4), 1007-1017 (2012)
Sun, H, Song, Y-Q, Chu, Y-M: Optimal two parameter bounds for the Seiffert mean. J. Appl. Math. 2013, Article ID 438971 (2013)
Gong, W-M, Song, Y-Q, Wang, M-K, Chu, Y-M: A sharp double inequality between Seiffert, arithmetic, and geometric means. Abstr. Appl. Anal. 2012, Article ID 684834 (2012)
Yang, Z-H, Song, Y-Q, Chu, Y-M: Monotonicity of the ratio of the power and second Seiffert means with applications. Abstr. Appl. Anal. 2014, Article ID 840130 (2014)
Sándor, J: On certain inequalities for means III. Arch. Math. 76(1), 30-40 (2001)
Jiang, W-D, Cao, J, Qi, F: Two sharp inequalities for bounds the Seiffert mean by the arithmetic, centroidal, and contraharmonic means. arXiv:1201.6432v1 [math.CA]
Sándor, J: Über zwei Mittel von Seiffert. Wurzel 36, 104-107 (2002)
Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 11401191, 11171307 and 61374086, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
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 is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
About this article
Cite this article
Chu, YM., Qian, WM., Wu, LM. et al. Optimal bounds for the first and second Seiffert means in terms of geometric, arithmetic and contraharmonic means. J Inequal Appl 2015, 44 (2015). https://doi.org/10.1186/s13660-015-0570-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0570-2