- Research
- Open access
- Published:
Optimal evaluation of a Toader-type mean by power mean
Journal of Inequalities and Applications volume 2015, Article number: 408 (2015)
Abstract
In this paper, we present the best possible parameters \(p, q\in\mathbb {R}\) such that the double inequality \(M_{p}(a,b)< T[A(a,b), Q(a,b)]< M_{q}(a,b)\) holds for all \(a, b>0\) with \(a\neq b\), and we get sharp bounds for the complete elliptic integral \(\mathcal{E}(t)=\int _{0}^{\pi/2}(1-t^{2}\sin^{2}\theta)^{1/2}\,d\theta\) of the second kind on the interval \((0, \sqrt{2}/2)\), where \(T(a,b)=\frac{2}{\pi }\int _{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}\,d\theta\), \(A(a,b)=(a+b)/2\), \(Q(a,b)=\sqrt{(a^{2}+b^{2})/2}\), \(M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}\) (\(r\neq0\)), and \(M_{0}(a,b)=\sqrt {ab}\) are the Toader, arithmetic, quadratic, and rth power means of a and b, respectively.
1 Introduction
For \(r\in\mathbb{R}\) and \(a,b>0\), the Toader mean \(T(a,b)\) (see [1]) and rth power mean \(M_{r}(a, b)\) are defined by
and
respectively.
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 bivariate means are a special case of the power mean, for example, \(H(a,b)=2ab/(a+b)=M_{-1}(a,b)\) is the harmonic mean, \(G(a,b)=\sqrt{ab}=M_{0}(a,b)\) is the geometric mean,
is the arithmetic mean, and
is the quadratic mean. The main properties of the power mean are given in [2]. The Toader mean \(T(a,b)\) has been well known in the mathematical literature for many years, it satisfies
where
stands for the symmetric complete elliptic integral of the second kind (see [3–5]), therefore it cannot be expressed in terms of the elementary transcendental functions.
Let \(r\in(0, 1)\), \(\mathcal{K}(r)=\int_{0}^{\pi /2}(1-r^{2}\sin ^{2}\theta)^{-1/2}\,d\theta\), and \(\mathcal{E}(r)=\int _{0}^{\pi /2}(1-r^{2}\sin^{2}\theta)^{1/2}\,d\theta\) be, respectively, the complete elliptic integrals of the first and second kind. Then \(\mathcal{K}(0^{+})=\mathcal{E}(0^{+})=\pi/2\), the Toader mean \(T(a,b)\) given in (1.1) can be expressed as
and \(\mathcal{K}(r)\) and \(\mathcal{E}(r)\) satisfy the derivatives formulas (see [6], Appendix E, p. 474-475)
Numerical computations show that
Recently, the power mean \(M_{r}(a,b)\) and Toader mean \(T(a,b)\) have been the subject of intensive research. In particular, many remarkable inequalities for both means can be found in the literature [7–18].
Vuorinen [19] conjectured that the inequality
holds for all \(a, b>0\) with \(a\neq b\). This conjecture was proved by Qiu and Shen [20], and Barnard et al. [21], respectively.
Alzer and Qiu [22] presented a best possible upper power mean bound for the Toader mean as follows:
for all \(a, b>0\) with \(a\neq b\).
Neuman [3], and Kazi and Neuman [4] proved that the inequalities
hold for all \(a, b>0\) with \(a\neq b\), where \(AGM(a,b)\) is the arithmetic-geometric mean of a and b.
Let \(\lambda, \mu, \alpha, \beta\in(1/2, 1)\). Then Chu et al. [23], and Hua and Qi [24] proved that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\) if and only if \(\lambda\leq3/4\), \(\mu\geq1/2+\sqrt{\pi(4-\pi)}/(2\pi)\), \(\alpha\leq1/2+\sqrt{3}/4\), and \(\beta\geq1/2+\sqrt{12/\pi-3}/2\), where \(C(a,b)=(a^{2}+b^{2})/(a+b)\) and \(\overline {C}(a,b)=2(a^{2}+ab+b^{2})/[3(a+b)]\) are, respectively, the contraharmonic and centroidal means of a and b.
In [25–29], the authors proved that the double inequalities
hold for all \(a, b>0\) with \(a\neq b\) if and only if \(\alpha_{1}\leq 1/2\), \(\beta_{1}\geq(4-\pi)/[(\sqrt{2}-1)\pi]\), \(\alpha_{2}\leq1/2\), \(\beta_{2}\geq4-2\log\pi/\log2\), \(\alpha _{3}\leq1/4\), \(\beta_{3}\geq4/\pi-1\), \(\alpha_{4}\leq\pi/2-1\), \(\beta_{4}\geq3/4\), \(\alpha_{5}\leq5/8\), \(\beta_{5}\geq2/\pi\), \(\alpha_{6}\leq1/8\), \(\beta_{6}\geq2/\pi-1/2\), \(\alpha_{7}\leq3/4\), \(\beta_{7}\geq12/\pi-3\), \(\alpha_{8}\leq\pi-3\), \(\beta_{8}\geq1/4\), \(\alpha_{9}\leq5/6\), \(\beta_{9}\geq2\sqrt {2}/\pi\), \(\alpha_{10}\leq0\), and \(\beta_{10}\geq1/6\).
The main purpose of this paper is to present the best possible parameters \(p, q\in\mathbb{R}\) 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 several lemmas which we present in this section.
Lemma 2.1
(See [30], Theorem 1.1)
The inequality \(\mathcal {E} [M_{p}(x, y) ]>M_{q}[\mathcal{E}(x), \mathcal{E}(y)]\) holds for all \(x, y\in(0, 1)\) if and only if
where \(q\rightarrow C(q)\) is a continuous function which satisfies \(C(q)=2\) for all \(q\leq5/2\) and \(C(q)<2\) for all \(q>5/2\).
Lemma 2.2
The double inequality
holds for all \(t\in(0, \sqrt{2}/2)\).
Proof
Let \(u=(1-t^{2})^{1/8}\). Then \(u\in(1/\sqrt[8]{2}, 1)\), \(t^{2}=1-u^{8}\), and the first inequality of (2.1) is equivalent to
for all \(u\in(1/\sqrt[8]{2}, 1)\).
We clearly see that (2.2) follows from
for all \(u\in(1/\sqrt[8]{2}, 1)\).
For the second inequality of (2.1), let \(v=\sqrt{1-t^{2}}\in(\sqrt {2}/2, 1)\), then it suffices to prove that
for all \(v\in(\sqrt{2}/2, 1)\).
We claim that
for all \(v\in(\sqrt{2}/2, 1)\).
Indeed, if \(v\in(\sqrt{2}/2, (\sqrt{6}-1)/2]\), then we clearly see that the function \(2-6v+3v^{2}+2v^{3}\) is strictly increasing on \((\sqrt{2}/2, (\sqrt{6}-1)/2]\), and (2.4) follows from
If \(v\in((\sqrt{6}-1)/2, 1)\), then (2.4) follows easily from
Therefore, inequality (2.3) follows from (2.4) and
for all \(v\in(\sqrt{2}/2, 1)\). □
Lemma 2.3
The inequality
holds for all \(t\in(0, 3/5)\).
Proof
Let
Then simple computations lead to
where
where
for \(t\in(0, 3/5)\).
From (2.9)-(2.11) we clearly see that \(f_{1}(t)\) is strictly decreasing on \((0, 3/5)\). Then (2.7) and (2.8) lead to the conclusion that there exists \(t_{0}\in(0, 3/5)\) such that \(f(t)\) is strictly increasing on \((0, t_{0}]\) and strictly decreasing on \([t_{0}, 3/5)\).
Therefore, Lemma 2.3 follows easily from (2.5) and (2.6) together with the piecewise monotonicity of \(f(t)\). □
Lemma 2.4
The inequality
holds for all \(t\in(0, 3/4)\).
Proof
It suffices to prove that the inequalities
and
hold for all \(t\in(0, 3/4)\).
Indeed, inequalities (2.12) and (2.13) follow easily from the identities
and
 □
Lemma 2.5
Let \(\lambda=2\log2/[2\log\pi-\log2-2\log\mathcal {E}(\sqrt{2}/2)]=1.3930\ldots\) and
Then \(g(t)>0\) for all \(t\in(0, 3/4)\).
Proof
It follows from \(t/\sqrt{1+t^{2}}\in(0, 3/5)\), \(\lambda<7/5\), Lemma 2.3, Lemma 2.4 and the monotonicity of \(M_{r}(1+t, 1-t)\) with respect to \(r\in\mathbb{R}\) that
for \(t\in(0, 3/4)\). Let
Then simple computations lead to
for \(t\in(0, 3/4)\).
From (2.18) and (2.19) we know that there exists \(t_{1}\in(0, 3/4)\) such that \(g^{\prime}_{1}(t)\) is strictly increasing on \((0, t_{1}]\) and strictly decreasing on \([t_{1}, 3/4)\). Then (2.17) leads to the conclusion that there exists \(t_{2}\in(0, 3/4)\) such that \(g_{1}(t)\) is strictly increasing on \((0, t_{2}]\) and strictly decreasing on \([t_{2}, 3/4)\).
Therefore, Lemma 2.5 follows from (2.14)-(2.16) and the piecewise monotonicity of \(g_{1}(t)\). □
Lemma 2.6
Let \(\lambda=2\log2/[2\log\pi-\log2-2\log\mathcal {E}(\sqrt{2}/2)]=1.3930\ldots\) . Then the function \(t^{-1}\mathcal {E}^{\lambda-1}(t)[\mathcal{E}(t)-\mathcal{K}(t)]\) is strictly decreasing on \((0, 1)\).
Proof
From Lemma 2.1 we clearly see that the inequality
holds for all \(x, y\in(0, 1)\) with \(x\neq y\).
It follows from the monotonicity of the function \(\mathcal{E}(t)\) and the power mean \(M_{p}(x, y)\) with respect to \(p\in\mathbb{R}\) together with \(\lambda>1\) that
for all \(x, y\in(0, 1)\) with \(x\neq y\).
Inequalities (2.20) and (2.21) lead to
for all \(x, y\in(0, 1)\) with \(x\neq y\), which implies that the function \(\mathcal{E}^{\lambda}(t)\) is strictly concave on \((0, 1)\).
Note that
 □
Therefore, Lemma 2.6 follows easily from (2.22) and the concavity of \(\mathcal{E}^{\lambda}(t)\) on \((0, 1)\).
Lemma 2.7
Let \(\lambda=2\log2/[2\log\pi-\log2-2\log\mathcal {E}(\sqrt{2}/2)]=1.3930\ldots\)
and
Then \(h_{1}(t)>0\) for \(t\in[3/5, 17/25)\) and \(h_{2}(t)>0\) for \(t\in [17/25, \sqrt{2}/2)\).
Proof
Simple computations lead to
From (2.24) and (2.25) together with Lemma 2.6 we clearly see that both \(h'_{1}(t)\) and \(h'_{2}(t)\) are strictly decreasing on \((0, \sqrt {2}/2)\). Then (2.26) leads to the conclusion that \(h_{1}(t)\) is strictly decreasing on \([3/5, 17/25]\) and \(h_{2}(t)\) is strictly decreasing on \([17/25, \sqrt{2}/2)\).
Therefore, Lemma 2.7 follows from (2.23) and the monotonicity of \(h_{1}(t)\) on \([3/5, 17/25]\) and \(h_{2}(t)\) on \([17/25, \sqrt{2}/2)\). □
Lemma 2.8
(See [18], Corollary 3.2)
The inequality
holds for all \(t\in(0, 1)\).
3 Main results
Theorem 3.1
Let \(\lambda=2\log2/[2\log\pi-\log2-2\log\mathcal {E}(\sqrt{2}/2)]=1.3930\ldots\) . Then the double inequality
holds for all \(a, b>0\) with \(a\neq b\) if and only if \(p\leq\lambda\) and \(q\geq3/2\).
Proof
Since the arithmetic mean \(A(a,b)\), quadratic mean \(Q(a,b)\), Toader mean \(T(a,b)\), and rth power mean \(M_{r}(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, we assume that \(a>b\). Let \(t=(a-b)/\sqrt{2(a^{2}+b^{2})}\). Then \(t\in(0, \sqrt{2}/2)\) and equations (1.2)-(1.5) lead to
We divide the proof into three cases.
Case 1 \(r\geq3/2\). Then it follows from (3.1) and (3.2) together with the monotonicity of \(M_{r}(a,b)\) with respect to r that
Therefore,
for all \(a, b>0\) with \(a\neq b\) follows from Lemmas 2.2 and 2.8 together with (3.3).
Case 2 \(r\leq\lambda\). Then equations (3.1) and (3.2) together with the monotonicity of \(M_{r}(a,b)\) with respect to r lead to
We divide the proof into two subcases.
Subcase 2.1 \(t\in(0, 3/5)\). Let \(u=t/\sqrt{1-t^{2}}\). Then \(u\in(0, 3/4)\) and (3.4) leads to
Therefore,
for \(0<\vert a-b\vert /\sqrt{2(a^{2}+b^{2})}<3/5\) with \(a\neq b\) follows from Lemma 2.5 and (3.5).
Subcase 2.2 \(t\in[3/5, \sqrt{2}/2)\). Let
It is easy to verify that
for all \(t\in(0, \sqrt{2}/2)\).
Equation (3.6) and inequality (3.7) lead to
and
Therefore,
for \(3/5\leq \vert a-b\vert /\sqrt{2(a^{2}+b^{2})}\) with \(a\neq b\) follows from Lemma 2.7, (3.4), (3.6), (3.8), and (3.9).
Case 3 \(\lambda< r<3/2\). On the one hand, equations (1.2) and (1.5) lead to
Inequality (3.10) implies that there exists \(\delta_{1}>0\) such that
for all \(a, b>0\) with \(a/b\in(0, \delta_{1})\).
On the other hand, by the Taylor expansion and let \(x>0\) and \(x\rightarrow0\), then equations (1.2) and (1.5) lead to
Equation (3.11) implies there exists \(\delta_{2}\in(0, 1)\) such that
for all \(a, b>0\) with \(a/b\in(1-\delta_{2}, 1)\). □
From Theorem 3.1 we get Corollary 3.2 immediately.
Corollary 3.2
Let \(\lambda=2\log2/[2\log\pi-\log2-2\log \mathcal {E}(\sqrt{2}/2)]=1.3930\ldots\) . Then the double inequality
holds for all \(t\in(0, \sqrt{2}/2)\) if and only if \(p\leq\lambda\) and \(q\geq3/2\).
References
Toader, G: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218(2), 358-368 (1998)
Bullen, PS, Mitrinović, DS, Vasić, PM: Means and Their Inequalities. Reidel, Dordrecht (1988)
Neuman, E: Bounds for symmetric elliptic integrals. J. Approx. Theory 122(2), 249-259 (2003)
Kazi, H, Neuman, E: Inequalities and bounds for elliptic integrals. J. Approx. Theory 146(2), 212-226 (2007)
Kazi, H, Neuman, E: Inequalities and bounds for elliptic integrals II. In: Special Functions and Orthogonal Polynomials. Contemp. Math., vol. 471, pp. 127-138. Am. Math. Soc., Providence, RI (2008)
Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)
Lin, TP: The power mean and the logarithmic mean. Am. Math. Mon. 81, 879-883 (1974)
Pittenger, AO: Inequalities between arithmetic and logarithmic means. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 678-715, 15-18 (1980)
Pittenger, AO: The symmetric, logarithmic and power means. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 678-715, 19-23 (1980)
Stolarsky, KB: The power and generalized logarithmic means. Am. Math. Mon. 87(7), 545-548 (1980)
Alzer, H: Ungleichungen für \((e/a)^{a}(b/e)^{b}\). Elem. Math. 40, 120-123 (1985)
Alzer, H: Ungleichungen für Mittelwerte. Arch. Math. 47(5), 422-426 (1986)
Burk, F: The geometric, logarithmic, and arithmetic mean inequality. Am. Math. Mon. 94(6), 527-528 (1987)
Alzer, H, Qiu, S-L: Inequalities for means in two variables. Arch. Math. 80(2), 201-215 (2003)
Costin, I, Toader, G: Optimal evaluations of some Seiffert-type means by power mean. Appl. Math. Comput. 219(9), 4745-4754 (2013)
Chu, Y-M, Wang, M-K, Qiu, S-L, Qiu, Y-F: Sharp generalized Seiffert mean bounds for Toader mean. Abstr. Appl. Anal. 2011, Article ID 605259 (2011)
Chu, Y-M, Wang, M-K: Inequalities between arithmetic-geometric, Gini, and Toader means. Abstr. Appl. Anal. 2012, Article ID 830585 (2012)
Chu, Y-M, Wang, M-K: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61(3-4), 223-229 (2012)
Vuorinen, M: Hypergeometric functions in geometric function theory. In: Special Functions and Differential Equations (Madras, 1977), pp. 119-126. Allied Publ., New Delhi (1998)
Qiu, S-L, Shen, J-M: On two problems concerning means. J. Hongzhou Inst. Electron. Eng. 17(3), 1-7 (1997) (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)
Chu, Y-M, Wang, M-K, Ma, X-Y: Sharp bounds for Toader mean in terms of contraharmonic mean with applications. J. Math. Inequal. 7(2), 161-166 (2012)
Hua, Y, Qi, F: A double inequality for bounding Toader mean by the centroidal mean. Proc. Indian Acad. Sci. Math. Sci. 124(4), 527-531 (2014)
Chu, Y-M, Wang, M-K, Qiu, S-L: Optimal combination bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41-51 (2012)
Song, Y-Q, Jiang, W-D, Chu, Y-M, Yan, D-D: Optimal bounds for Toader mean in terms of arithmetic and contraharmonic means. J. Math. Inequal. 7(4), 751-757 (2013)
Li, W-H, Zheng, M-M: Some inequalities for bounding Toader mean. J. Funct. Spaces Appl. 2013, Article ID 394194 (2013)
Hua, Y, Qi, F: The best bounds for Toader mean in terms of the centroidal and arithmetic means. Filomat 28(4), 775-780 (2014)
Sun, H, Chu, Y-M: Bounds for Toader mean by quadratic and harmonic means. Acta Math. Sci. 35A(1), 36-42 (2015) (in Chinese)
Chu, Y-M, Wang, M-K, Jiang, Y-P, Qiu, S-L: Concavity of the complete elliptic integrals of the second kind with respect to Hölder means. J. Math. Anal. Appl. 395(2), 637-642 (2012)
Acknowledgements
This research was supported by the Natural Science Foundation of China under Grants 11301127, 11371125, 11401191, and 61374086, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004 and the Natural Science Foundation of Hunan Province under Grant 12C0577.
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
Song, YQ., Zhao, TH., Chu, YM. et al. Optimal evaluation of a Toader-type mean by power mean. J Inequal Appl 2015, 408 (2015). https://doi.org/10.1186/s13660-015-0927-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-015-0927-6