Abstract
In this paper, we find the greatest values \(\alpha_{1},\alpha_{2}\) and the smallest values \(\beta_{1},\beta_{2}\) such that the double inequalities \(L_{\alpha_{1}}(a,b)<\operatorname{AG}(a,b)<L_{\beta_{1}}(a,b)\) and \(L_{\alpha_{2}}(a,b)< T(a,b)< L_{\beta_{2}}(a,b)\) hold for all \(a, b>0\) with \(a\neq b\), where \(\operatorname{AG}(a,b)\), \(T(a,b)\) and \(L_{p}(a,b)\) are the arithmetic-geometric, Toader and generalized logarithmic means of two positive numbers a and b, respectively.
Similar content being viewed by others
1 Introduction
For \(p \in\mathbb{R}\), the pth generalized logarithmic mean \(L_{p}(a,b)\) [1] of two positive numbers a and b is defined by
It is well known that \(L_{p}(a,b)\) is continuous and strictly increasing with respect to \(p\in\mathbb{R}\) for fixed \(a,b > 0\) with \(a\neq b\). Many remarkable inequalities for the generalized logarithmic mean can be found in the literature [2–17].
The classical arithmetic-geometric mean \(\operatorname{AG}(a,b)\) of two positive numbers a and b is defined by starting with \(a_{0}=a,b_{0}=b\) and then iterating
for \(n\in\mathbb{N}\) until two sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) converge to the same number.
The well-known Gauss identity [18] shows that
for \(r\in(0,1)\), where \(\mathcal{K}(r)=\int^{\pi/2}_{0}(1-r^{2}\sin ^{2}t)^{-1/2}\,dt\), \(r\in[0,1)\), is the complete elliptic integral of the first kind.
In [19], the Toader mean \(T(a,b)\) of two positive numbers a and b was given by
where \(\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta )^{1/2}\,d\theta\), \(r\in[0,1]\) is the complete elliptic integral of the second kind.
Recently, the bounds for the arithmetic-geometric mean \(\operatorname{AG}(a,b)\) and Toader mean \(T(a,b)\) have attracted the attention of many mathematicians. The double inequality
holds for all \(a,b>0\) with \(a\neq b\). The left inequality of (1.5) was first proposed by Carlson and Vuorinen [20] and also was proved by different methods in [21–23]. Vamanamurthy and Vuorinen [24] proved that \(\operatorname{AG}(a,b)<(\pi/2)L(a,b)\) for all \(a,b>0\) with \(a\neq b\). The second inequality of (1.5) was proved by Borwein and Borwein [25] and Yang [23].
Vuorinen [26] conjectured that
for all \(a,b>0\) with \(a\neq b\), where \(M_{p}(a,b)=[(a^{p}+b^{p})/2]^{1/p}\) (\(p\neq0\)) and \(M_{0}(a,b)=\sqrt{ab}\) is the power mean of order p. This conjecture was proved by Qiu and Shen [27] and Barnard et al. [28].
In [29], Alzer and Qiu presented a best possible upper power mean bound for the Toader mean as follows:
for all \(a,b>0\) with \(a\neq b\).
In [30–32], the authors proved that
for all \(a,b>0\) with \(a\neq b\), where \(\widehat {L}_{p}(a,b)=(a^{p+1}+b^{p+1})/(a^{p}+b^{p})\) denotes the pth Lehmer mean and \(\widehat{S}_{p}(a,b)\) is the generalized Seiffert mean given by \(\widehat{S}_{p}(a,b)=p(a-b)/\arctan[2p(a-b)/(a+b)]\) \((0< p\leq1, a\neq b)\), \(\widehat{S}_{0}(a,b)=(a+b)/2 (a\neq b)\) and \(\widehat{S}_{p}(a,a)=a\).
Very recently, Chu and Wang [33] proved that
for all \(a,b>0\) with \(a\neq b\) if and only if \(p_{1}\leq1/2,q_{1}\geq1\) and \(p_{2}\leq1,q_{2}\geq3/2\). Here the pth Gini mean of two positive numbers a and b is defined by
The main purpose of this paper is to find the greatest values \(\alpha _{1},\alpha_{2}\) and the smallest values \(\beta_{1},\beta_{2}\) such that the double inequalities \(L_{\alpha_{1}}(a,b)<\operatorname{AG}(a,b)<L_{\beta_{1}}(a,b)\) and \(L_{\alpha_{2}}(a,b)< T(a,b)< L_{\beta_{2}}(a,b)\) hold for all \(a, b>0\) with \(a\neq b\) and give some new bounds for the complete elliptic integrals.
2 Basic knowledge and lemmas
In order to prove our main results, we need several formulas and lemmas, which we present in this section.
For \(r\in(0, 1)\) and \(r'=\sqrt{1-r^{2}}\), the well-known complete elliptic integrals of the first and second kinds are defined by
and
respectively, and the following formulas were presented in [18], Appendix E, pp.474-475:
In what follows, four special values \(\mathcal{E}(\sqrt {2}/2),\mathcal{K}(\sqrt{2}/2)\) and \(\mathcal{E}(0.9),\mathcal {K}(0.9)\) will be used. By numerical computations, these are given by
Lemma 2.1
See [18], Theorem 1.25
For \(-\infty< a < b <\infty\), let \(f,g:[a, b]\rightarrow\mathbb{R}\) be continuous on \([a, b]\) and be differentiable on \((a, b)\), let \(g'(x)\neq0\) on \((a, b)\). If \(f'(x)/g'(x)\) is increasing (decreasing) on \((a, b)\), then so are
If \(f '(x)/g'(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2
(1) The function \(r\rightarrow(\mathcal{E}-r^{\prime 2}\mathcal{K})/r^{2}\) is strictly increasing from \((0, 1)\) onto \((\pi/4, 1)\);
(2) The function \(r\rightarrow2\mathcal{E}-r^{\prime 2}\mathcal{K}\) is increasing and log-convex from \((0,1)\) onto \((\pi/2,2)\);
(3) The function \(\mathcal{K}/\log(e^{2}/r')\) is strictly increasing from \((0,1)\) onto \((\pi/4, 1)\);
(4) The function \((\mathcal{K}-\mathcal{E})/r^{2}\) is strictly increasing on \((0,1)\); in particular, \(\mathcal{K}-\mathcal{E}>(\pi /4)r^{2}\) for all \(r\in(0,1)\).
Proof
Parts (1) and (2) follow from [18], Theorem 3.21(1) and Exercise 3.43(13). □
Lemma 2.3
The equation
has a unique solution \(p=p_{0}=3.15295\cdots\) .
Proof
Let
It is easy to verify that the function φ is continuous and strictly decreasing from \((-1,+\infty)\) onto \((1,+\infty)\). Therefore, Lemma 2.3 easily follows from the continuity and monotonicity of φ together with the facts that \(\varphi(3.15295)=6.14999\times10^{-7}\) and \(\varphi (3.15296)=-4.35155\times10^{-7}\). □
Lemma 2.4
The function
is strictly increasing from \((0,1)\) onto \((1/64,4/\pi-5/4)\).
Proof
Let \(f_{1}(r)=2(2\mathcal{E}-r^{\prime 2}\mathcal{K})/\pi-1-r^{2}/4\) and \(\widehat{f}_{1}(r)=r^{4}\), then \(f_{1}(0)=\widehat{f}_{1}(0)=0\) and \(f(r)=f_{1}(r)/\widehat{f}_{1}(r)\).
A simple calculation yields
Following from Lemma 2.2(1) and (2.9) together with the monotonicity of \(1/r^{\prime 2}\), we clearly see that \(f'_{3}(r)/\widehat{f}'_{3}(r)\) is strictly increasing on \((0,1)\). Equations (2.5)-(2.8) and Lemma 2.1 lead to the conclusion that \(f(r)\) is strictly increasing on \((0,1)\).
Therefore, Lemma 2.4 follows from the monotonicity of \(f(r)\) together with the facts that \(f(0^{+})=1/64\) and \(f(1^{-})=4/\pi-5/4\). □
The following double inequalities can be obtained from Lemma 2.4 immediately.
Corollary 2.5
Inequalities
hold for \(0< r<1\).
Lemma 2.6
The inequality
holds for \(0< r<1\).
Proof
In order to prove inequality (2.10), it suffices to prove that
for \(0< r<1\), where
Observe that
we conclude, from (2.12) and (2.13), that there exists \(r_{0}\in (0.56,0.57)\) such that \(g_{1}(r)>0\) for \(r\in(0,r_{0})\) and \(g_{1}(r)<0\) for \(r\in(r_{0},1)\).
In order to prove (2.11), we divide it into two cases.
Case A \(r\in[r_{0},1)\). In this case, we clearly see that \(g_{1}(r)\leq0\) and \(g_{2}(r)>0\). This implies that \(g(r)=g_{1}(r)-g_{2}(r)<0\).
Case B \(r\in(0,r_{0})\). In this case, \(g_{1}(r)>0\). Let \(g_{3}(r)=2-7r^{2}+\frac{35}{4}r^{4}-6r^{6}\), the difference between \(g_{1}(r)\) and \(g_{3}(r)\) yields
We know from (2.14) that \(g_{3}(r)>g_{1}(r)>0\). Moreover,
this in conjunction with \(g_{3}(r)>0\) implies that
Therefore, we clearly see that \(g(r)=[g_{1}(r)-g_{3}(r)]+[g_{3}(r)-g_{2}(r)]<0\) from (2.14) and (2.15). □
Lemma 2.7
Let \(\eta(r)=[(1+r)^{p_{0}+1}-(1-r)^{p_{0}+1}]/r\) and \(\omega (r)=[(1-r)^{p_{0}}(1+p_{0}r)-(1+r)^{p_{0}}(1-p_{0}r)]/r^{2}\), then the functions \(\eta(r)\) and \(\omega(r)\) both are strictly increasing on \((0,1)\).
Proof
We assume that
then \(\eta(r)=\eta_{1}(r)/\eta_{2}(r)\) and \(\omega(r)=\omega _{1}(r)/\omega_{2}(r)\).
A simple calculation yields
Lemma 2.1 and (2.16)-(2.18) lead to the conclusion that \(\eta(r)\) and \(\omega(r)\) are strictly increasing on \((0,1)\). □
Lemma 2.8
Let
then \(\phi_{p}(r)>0\) for \(0< r<1\) if and only if \(p\leq5/2\); \(\phi _{p}(r)<0\) for \(0< r<1\) if and only if \(p\geq p_{0}\).
Proof
It is well known that \(L_{p}(a,b)\) is strictly increasing with respect to \(p\in\mathbb{R}\) for fixed \(a,b>0\) with \(a\neq b\), then \(\phi_{p}(r)\) is strictly decreasing with respect to \(p\in\mathbb{R}\). In order to prove Lemma 2.8, we divide it into three cases.
Case 1 \(p=5/2\).
From Corollary 2.5 and Lemma 2.6, we clearly see that
for \(0< r<1\).
Case 2 \(p=p_{0}\).
We divide it into two subcases.
Subcase A \(\phi_{p_{0}}(r)<0\) for \(r\in(0,0.9)\).
Since \(\phi_{p}(r)\) is strictly decreasing with respect to \(p\in\mathbb {R}\), we clearly see that \(\phi_{p_{0}}(r)<\phi_{3}(r)\). It suffices to prove that \(\phi_{3}(r)<0\) for \(r\in(0,0.9)\).
For \(r\in(0,\sqrt{2}/2]\), it follows from Corollary 2.5 that
where the first inequality easily follows from
For \(r\in(\sqrt{2}/2,0.9)\), taking the derivative of \(\phi_{3}(r)\) yields
where
From Lemma 2.2(4), we clearly see that
for \(r\in(0,1)\) and
for \(r\in(\sqrt{2}/2,0.9)\). Equations (2.19)-(2.21) lead to the conclusion that \(\phi'_{3}(r)\) is strictly increasing on \((\sqrt {2}/2,0.9)\). This in conjunction with (2.3) implies that
for \(r\in(\sqrt{2}/2,9/10)\). Therefore, from (2.22) we clearly see that \(\phi_{3}(r)\) is strictly increasing on \((\sqrt{2}/2,0.9)\). This in conjunction with (2.4) yields \(\phi_{3}(r)<\phi _{3}(0.9)=-0.002687\cdots<0\) for \(r\in(\sqrt{2}/2,0.9)\).
Subcase B \(\phi_{p_{0}}(r)<0\) for \(r\in[0.9,1)\).
For \(0.9\leq r<1\), taking the derivation of \(\phi_{p_{0}}(r)\) yields
where \(\omega(r)\) and \(\eta(r)\) are defined as in Lemma 2.7. From Lemma 2.2(1), we clearly see that \((\mathcal{E}-r^{\prime 2}\mathcal{K})/r\) is strictly increasing on \((0,1)\). Lemma 2.7 and (2.4), (2.20) lead to the conclusion that
for \(0.9\leq r<1\).
Therefore, it follows from the monotonicity of \(\phi'_{p_{0}}(r)\) on \((9/10,1)\) that \(\phi_{p_{0}}(r)<\phi_{p_{0}}(1)=2 [2/\pi -1/(1+p_{0})^{1/p_{0}} ]=0\) for \(0< r<1\).
Case 3 \(5/2< p< p_{0}\).
Taking the Taylor series of \(\phi_{p}(r)\) at \(r=0\) yields
From (2.24) we clearly see that there exists a sufficiently small \(\delta_{1}>0\) such that \(\phi_{p}(r)<0\) for \(r\in(0,\delta_{1})\) if \(p>5/2\). If \(p< p_{0}\), then \(\phi_{p}(1)=2[2/\pi-1/(1+p)^{1/p}]>0\). By the continuity of \(\phi_{p}(r)\) with respect to r, there exists a sufficiently small \(\delta_{2}>0\) such that \(\phi_{p}(r)>0\) for \(r\in (\delta_{2},1)\). □
3 Main results
Theorem 3.1
Inequality \(L_{-1}(a,b)<\operatorname{AG}(a,b)<L_{-1/2}(a,b)\) holds for all \(a, b>0\) with \(a\neq b\), where \(L_{-1}(a,b)\) and \(L_{-1/2}(a,b)\) are the best possible lower and upper generalized logarithmic mean bounds for the arithmetic-geometric mean \(\operatorname{AG}(a,b)\), respectively.
Proof
Firstly, from (1.5) we clearly see that \(L_{-1}(a,b)<\operatorname{AG}(a,b)\) for all \(a, b>0\) with \(a\neq b\).
Next, we prove that \(\operatorname{AG}(a,b)< L_{-1/2}(a,b)\) for all \(a, b>0\) with \(a\neq b\). Since \(\operatorname{AG}(a,b)\) and \(L_{p}(a,b)\) are symmetric and homogeneous of degree 1, without loss of generality, it suffices to give an assumption that \(a=1 > b\). Let \(t=b\in(0,1)\), \(r=(1-t)/(1+t)\), then (1.1) and (1.3) lead to
where
We can rewrite \(h(r)\) as
where \(\lambda(r')=(1+r')\log(e^{2}/r')\).
A simple calculation yields
Equations (3.3)-(3.5) lead to the conclusion that \(\lambda(r')\) is strictly decreasing on \((0,1)\) with respect to \(r'\). Moreover, the function \(r'=\sqrt{1-r^{2}}\) is strictly decreasing on \((0,1)\). Hence the function \(\lambda(r')\) is strictly increasing on \((0,1)\) with respect to r. It follows from (3.2) and Lemma 2.2(3) that \(h(r)\) is strictly decreasing on \((0,1)\). This implies that \(h(r)<0\) for \(0< r<1\) together with \(h(0)=0\).
Therefore, \(\operatorname{AG}(a,b)< L_{-1/2}(a,b)\) for all \(a, b>0\) with \(a\neq b\) follows from (3.1) and \(h(r)<0\).
Finally, we prove that \(L_{-1}(a,b)\) and \(L_{-1/2}(a,b)\) are the best possible lower and upper generalized logarithmic mean bounds for the arithmetic-geometric mean \(\operatorname{AG}(a,b)\).
For any \(0<\varepsilon<1/2\) and \(0< x<1\), it follows from (1.1) and (1.3) that
and making use of the Taylor expansion as \(x\rightarrow0\), one has
Equations (3.6) and (3.7) imply that for any \(0<\varepsilon<1/2\) there exist \(\delta_{1}=\delta_{1}(\varepsilon)\in(0,1)\) and \(\delta _{2}=\delta_{2}(\varepsilon)\in(0,1)\) such that \(\operatorname{AG}(1,x)< L_{-1+\varepsilon}(1,x)\) for \(x\in(0,\delta_{1})\) and \(\operatorname{AG}(1,1-x)>L_{-1/2-\varepsilon}(1,1-x)\) for \(x\in(0,\delta_{2})\). □
Theorem 3.2
Inequality \(L_{5/2}(a,b)< T(a,b)< L_{p_{0}}(a,b)\) holds for all \(a, b>0\) with \(a\neq b\), where \(p_{0}\) is defined as in Lemma 2.3 and \(L_{5/2}(a,b), L_{p_{0}}(a,b)\) are the best possible lower and upper generalized logarithmic mean bounds for the Toader mean \(T(a,b)\), respectively.
Proof
From (1.1) and (1.4) we clearly see that both \(T(a,b)\) and \(L_{p}(a,b)\) are symmetric and homogeneous of degree 1. Without loss of generality, we assume that \(a=1 > b\). Let \(t=b\in(0,1)\), \(r=(1-t)/(1+t)\), then from (1.1) and (1.4) together with (2.2) we have
where \(\phi_{p}(r)\) is defined as in Lemma 2.8.
4 Corollaries and remarks
From Theorem 3.1 we get a lower bound for the complete elliptic integral of the first kind \(\mathcal{K}(r)\) as follows.
Corollary 4.1
Inequality
holds for all \(r\in(0,1)\).
Remark 4.1
We define \(H(r)=2\pi[1+\sqrt{1-r^{2}}-2(1-r^{2})^{1/4}]/ (1-\sqrt{1-r^{2}} )^{2}\). Computational and numerical experiments show that the lower bound in (4.1) can be regarded as an approximation of \(\mathcal{K}(r)\) for some \(r\in(0,1)\), refer to Table 1 for numerical values.
Theorem 3.2 enables us to give new bounds for the complete elliptic integrals of the second kind \(\mathcal{E}(r)\).
Corollary 4.2
Inequality
holds for all \(r\in(0,1)\), where \(p_{0}=3.15295\cdots\) is defined as in Lemma 2.3.
References
Stolarsky, KB: Generalizations of the logarithmic mean. Math. Mag. 48, 87-92 (1975)
Chu, YM, Hou, SW, Gong, WM: Inequalities between logarithmic, harmonic, arithmetic and centroidal means. J. Math. Anal. 2(2), 1-5 (2011)
Chu, YM, Wang, MK: Optimal inequalities between harmonic, geometric, logarithmic, and arithmetic-geometric means. J. Appl. Math. 2011, Article ID 618929 (2011)
Qiu, YF, Wang, MK, Chu, YM, Wang, GD: Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean. J. Math. Inequal. 5(3), 301-306 (2011)
Chu, YM, Xia, WF: Two optimal double inequalities between power mean and logarithmic mean. Comput. Math. Appl. 60(1), 83-89 (2010)
Xia, WF, Chu, YM, Wang, GD: The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means. Abstr. Appl. Anal. 2010, Article ID 604804 (2010)
Xia, WF, Chu, YM: Optimal inequalities related to the logarithmic, identric, arithmetic and harmonic means. Rev. Anal. Numér. Théor. Approx. 39(2), 176-183 (2010)
Chu, YM, Xia, WF: Inequalities for generalized logarithmic means. J. Inequal. Appl. 2009, Article ID 763252 (2009)
Shi, HN, Wu, SH: Refinement of an inequality for the generalized logarithmic mean. Chin. Q. J. Math. 23(4), 594-599 (2008)
Qi, F, Chen, SX, Chen, CP: Monotonicity of ratio between the generalized logarithmic means. Math. Inequal. Appl. 10(3), 559-564 (2007)
Li, X, Chen, CP, Qi, F: Monotonicity result for generalized logarithmic means. Tamkang J. Math. 38(2), 177-181 (2007)
Chen, CP, Qi, F: Monotonicity properties for generalized logarithmic means. Aust. J. Math. Anal. Appl. 1(2), Article 2 (2004)
Mond, B, Pearce, CEM, Pečarić, J: The logarithmic mean is a mean. Math. Commun. 2(1), 35-39 (1997)
Kahlig, P, Matkowski, J: Functional equations involving the logarithmic mean. Z. Angew. Math. Mech. 76(7), 385-390 (1996)
Pittenger, AO: The logarithmic mean in n variables. Am. Math. Mon. 92(2), 99-104 (1985)
Stolarsky, KB: The power and generalized logarithmic means. Am. Math. Mon. 87(7), 545-548 (1980)
Lin, TP: The power mean and the logarithmic mean. Am. Math. Mon. 81, 879-883 (1974)
Anderson, GD, Vamanamurthy, MK, Vuorinen, MK: Conformal Invariants, Inequalities, and Quasiconformal Maps. Wiley, New York (1997)
Toader, G: Some mean values related to the arithmetic-geometric mean. J. Math. Anal. Appl. 218, 358-368 (1998)
Carlson, BC, Vuorinen, MK: Inequality of the AGM and the logarithmic mean. SIAM Rev. 33, 653-654 (1991)
Sándor, J: On certain inequalities for means. J. Math. Anal. Appl. 189, 602-606 (1995)
Neuman, E, Sándor, J: On certain means of two arguments and their extensions. Int. J. Math. Math. Sci. 16, 981-993 (2003)
Yang, ZH: A new proof of inequalities for Gauss compound mean. Int. J. Math. Anal. 4, 1013-1018 (2010)
Vamanamurthy, MK, Vuorinen, MK: Inequalities for means. J. Math. Anal. Appl. 183, 155-166 (1994)
Borwein, JM, Borwein, PB: Inequalities for compound mean iterations with logarithmic asymptotes. J. Math. Anal. Appl. 177, 572-582 (1993)
Vuorinen, MK: Hypergeometric functions in geometric function theory. In: Special Functions and Differential Equations, pp. 119-126. Allied Publ., New Delhi (1998)
Qiu, SL, Shen, JM: On two problems concerning means. J. Hangzhou Inst. Electron. Eng. 17, 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, 693-699 (2000)
Alzer, H, Qiu, SL: Monotonicity theorems and inequalities for the complete elliptic integrals. J. Comput. Appl. Math. 172, 289-312 (2004)
Chu, YM, Wang, MK: Optimal Lehmer mean bounds for the Toader mean. Results Math. 61, 223-229 (2012)
Guo, BN, Qi, F: Some bounds for the complete elliptic integrals of the first and second kinds. Math. Inequal. Appl. 14, 323-334 (2011)
Chu, YM, Wang, MK, Qiu, SL, Qiu, YF: Sharp generalized Seiffert mean bounds for Toader mean. Abstr. Appl. Anal. 2011, Article ID 605259 (2011)
Chu, YM, Wang, MK: Inequalities between arithmetic-geometric, Gini, and Toader means. Abstr. Appl. Anal. 2012, Article ID 830585 (2012)
Acknowledgements
The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions. This research was supported by the National Natural Science Foundation of China (Grant no. 11301127), the general project Foundation of the Department of Education of Hunan Province (Grant no. 16C0265) and the scientific research Foundation for Young teachers in Hunan University of Finance and Economics (Grant no. Q201501).
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.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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
Ding, Q., Zhao, T. Optimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean. J Inequal Appl 2017, 102 (2017). https://doi.org/10.1186/s13660-017-1365-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-017-1365-4