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Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters

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...

Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

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

Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral

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.

Optimal Bounds for Gaussian Arithmetic-Geometric Mean with Applications to Complete Elliptic Integral

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.

Sharp One-Parameter Mean Bounds for Yang Mean

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

Optimal bounds for two Sándor-type means in terms of power means

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...

Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

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 .

Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means

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 .

Sharp bounds for Toader mean in terms of arithmetic, quadratic, and Neuman means

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...

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

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...

Sharp Bounds for Toader Mean in terms of Arithmetic and Second Contraharmonic Means

: 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

Optimal bounds for Neuman means in terms of geometric, arithmetic and quadratic means

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.

Refinements of Bounds for Neuman Means

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

Refinements of Bounds for Neuman Means

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

Sharp bounds for Neuman means in terms of one-parameter family of bivariate means

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.

Sharp power-type Heronian mean bounds for the Sándor and Yang means

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...

Optimal bounds for the first and second Seiffert means in terms of geometric, arithmetic and contraharmonic means

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...

Bounds for the Arithmetic Mean in Terms of the Neuman-Sándor and Other Bivariate Means

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.

On Certain Inequalities for Neuman-Sándor Mean

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

On Certain Inequalities for Neuman-Sándor Mean

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