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In the article, we present the best possible parameters λ = λ ( p ) $\lambda=\lambda (p)$ and μ = μ ( p ) $\mu=\mu(p)$ on the interval [ 0 , 1 / 2 ] $[0, 1/2]$ such that the double inequality G p [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] A 1 − p ( a , b ) < E ( a , b ) < G p [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] A 1 − p ( a , b ) $$\begin{aligned}& G^{p}\bigl[\lambda a+(1...

2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
Received 23 January 2016; Revised 16 March 2016; Accepted 28 March 2016
Academic Editor: Kishin Sadarangani
Copyright © 2016 **Wei**-**Mao** ... **Qian** and Yu-Ming Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the

We present the best possible parameters and such that the double inequalities , , hold for all with , where , , and are the arithmetic, quadratic, and Gauss arithmetic-geometric means of and , respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.

We present the best possible parameters and such that the double inequalities , , hold for all with , where , , and are the arithmetic, quadratic, and Gauss arithmetic-geometric means of and , respectively. As applications, we find several new bounds for the complete elliptic integrals of the first and second kind.

Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China
Received 19 July 2015; Accepted 9 February 2016
Academic Editor: Yann Favennec
Copyright © 2016 **Wei**-**Mao** **Qian** et al. This is

In the article, we prove that the double inequalities M α ( a , b ) < S Q A ( a , b ) < M β ( a , b ) and M λ ( a , b ) < S A Q ( a , b ) < M μ ( a , b ) hold for all a , b > 0 with a ≠ b if and only if α ≤ log 2 / [ 1 + log 2 − 2 log ( 1 + 2 ) ] = 1.5517 … , β ≥ 5 / 3 , λ ≤ 4 log 2 / [ 4 + 2 log 2 − π ] = 1.2351 … and μ ≥ 4 / 3 , where S Q A ( a , b ) = A ( a , b ) e Q ( a , b...

We present the best possible parameters and such that the double inequalities and hold for all with and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here, , , , and are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of and , and .

We present the best possible parameters and such that the double inequalities and hold for all with and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here, , , , and are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of and , and .

In this paper, we present the best possible parameters α , β ∈ R and λ , μ ∈ ( 1 / 2 , 1 ) such that the double inequalities α N A Q ( a , b ) + ( 1 − α ) A ( a , b ) < T ∗ ( a , b ) < β N A Q ( a , b ) + ( 1 − β ) A ( a , b ) , Q [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] < T ∗ ( a , b ) < Q [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] hold for all a , b > 0 with a ≠ b , where T...

In the article, we present the best possible parameters α 1 , α 2 , β 1 , β 2 ∈ ( 0 , 1 ) and α 3 , α 4 , β 3 , β 4 ∈ ( 0 , 1 / 2 ) such that the double inequalities α 1 A ( a , b ) + ( 1 − α 1 ) H ( a , b ) < X ( a , b ) < β 1 A ( a , b ) + ( 1 − β 1 ) H ( a , b ) , α 2 A ( a , b ) + ( 1 − α 2 ) G ( a , b ) < X ( a , b ) < β 2 A ( a , b ) + ( 1 − β 2 ) G ( a , b ) , H [ α 3 a...

: Lars E. Persson
Copyright © 2015 **Wei**-**Mao** **Qian** et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and

In this paper, we present sharp bounds for the two Neuman means SHA and SCA derived from the Schwab-Borchardt mean in terms of convex combinations of either the weighted arithmetic and geometric means or the weighted arithmetic and quadratic means, and the mean generated either by the geometric or by the quadratic mean.MSC: 26E60.

and **Wei**-**Mao** **Qian**. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the

and **Wei**-**Mao** **Qian**. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the

We present the best possible parameters p1,p2,p3,p4,q1,q2,q3,q4∈[0,1] such that the double inequalities Gp1(a,b)<SHA(a,b)<Gq1(a,b), Qp2(a,b)<SCA(a,b)<Qq2(a,b), Hp3(a,b)<SAH(a,b)<Hq3(a,b), Cp4(a,b)<SAC(a,b)<Cq4(a,b) hold for all a,b>0 with a≠b, where SHA, SCA, SAH, SAC are the Neuman means, and Gp, Qp, Hp, Cp are the one-parameter means.MSC: 26E60.

We prove that the double inequalities H α ( a , b ) < X ( a , b ) < H β ( a , b ) and H λ ( a , b ) < U ( a , b ) < H μ ( a , b ) hold for all a , b > 0 with a ≠ b if and only if α ≤ 1 / 2 , β ≥ log 3 / ( 1 + log 2 ) = 0.6488 ⋯ , λ ≤ 2 log 3 / ( 2 log π − log 2 ) = 1.3764 ⋯ , and μ ≥ 2 , where H p ( a , b ) , X ( a , b ) , and U ( a , b ) are, respectively, the pth power-type...

In this paper, we find the greatest values α, λ and the least values β, μ such that the double inequalities α [ G ( a , b ) / 3 + 2 A ( a , b ) / 3 ] + ( 1 − α ) G 1 / 3 ( a , b ) A 2 / 3 ( a , b ) < P ( a , b ) < β [ G ( a , b ) / 3 + 2 A ( a , b ) / 3 ] + ( 1 − β ) G 1 / 3 ( a , b ) A 2 / 3 ( a , b ) and λ [ C ( a , b ) / 3 + 2 A ( a , b ) / 3 ] + ( 1 − λ ) C 1 / 3 ( a , b ) A...

We present the largest values , , and and the smallest values , , and such that the double inequalities , , and hold for all with , where , , , and denote the Neuman-Sándor, arithmetic, Heronian, harmonic, and harmonic root-square means of and , respectively.

Computation Sciences, Hunan City University, Yiyang 413000, China
Received 2 March 2013; Accepted 14 April 2013
Academic Editor: Josef Diblík
Copyright © 2013 **Wei**-**Mao** **Qian** and Yu-Ming Chu. This is an open

We present several new sharp bounds for Neuman-Sándor mean in terms of arithmetic, centroidal, quadratic, harmonic root square, and contraharmonic means.