Sharp Geometric Mean Bounds for Neuman Means

Yan Zhang,Yu-Ming Chu,Yun-Liang Jiang

doi:10.1155/2014/949815

Yan Zhang, Yu-Ming Chu + Show 1 more

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https://doi.org/10.1155/2014/949815

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Abstract

We find the best possible constantsα1,α2,β1,β2∈[0,1/2]andα3,α4,β3,β4∈[1/2,1]such that the double inequalitiesG(α1a+(1-α1)b,α1b+(1-α1)a)<NAG(a,b)<G(β1a+(1-β1)b,β1b+(1-β1)a),G(α2a+(1-α2)b,α2b+(1-α2)a)<NGA(a,b)<G(β2a+(1-β2)b,β2b+(1-β2)a),Q(α3a+(1-α3)b,α3b+(1-α3)a)<NQA(a,b)<Q(β3a+(1-β3)b,β3b+(1-β3)a),Q(α4a+(1-α4)b,α4b+(1-α4)a)<NAQ(a,b)<Q(β4a+(1-β4)b,β4b+(1-β4)a)hold for alla,b>0witha≠b, whereG,A, andQare, respectively, the geometric, arithmetic, and quadratic means andNAG,NGA,NQA, andNAQare the Neuman means.

Highlights

B > 0 with a ≠ b, the Schwab-Borchardt mean SB(a, b) [1, 2] of a and b is given by {{{{ √b2 cos−1 − a2 (a/b), a < b, SB (a, b) = {{{{ {√a2 − b2 cosh−1 (a/b), a > b, (1)where cos−1(x) and cosh−1(x) = log(x + √x2 − 1) are the inverse cosine and inverse hyperbolic cosine functions, respectively
The Schwab-Borchardt mean has been the subject of intensive research
Many remarkable inequalities for Schwab-Borchardt mean and its generated means can be found in the literature [1,2,3,4,5,6]

Summary

Introduction

We find the best possible constants α1, α2, β1, β2 ∈ [0, 1/2] and α3, α4, β3, β4 ∈ [1/2, 1] such that the double inequalities G(α1a + (1 − α1)b, α1b + (1 − α1)a) < NAG(a, b) < G(β1a + (1 − β1)b, β1b + (1 − β1)a), G(α2a + (1 − α2)b, α2b + (1 − α2)a) < NGA(a, b) < G(β2a + (1 − β2)b, β2b + (1 − β2)a), Q(α3a + (1 − α3)b, α3b + (1 − α3)a) < NQA(a, b) < Q(β3a + (1 − β3)b, β3b + (1 − β3)a), Q(α4a + (1 − α4)b, α4b + (1 − α4)a) < NAQ(a, b) < Q(β4a + (1 − β4)b, β4b + (1 − β4)a) hold for all a, b > 0 with a ≠ b, where G, A, and Q are, respectively, the geometric, arithmetic, and quadratic means and NAG, NGA, NQA, and NAQ are the Neuman means. Let NAG(a, b) = N(A(a, b), G(a, b)), NGA(a, b) = N(G(a, b), A(a, b)), NQA(a, b) = N(Q(a, b), A(a, b)), and NAQ(a, b) = N(A(a, b), Q(a, b)) be the Neuman means, where G(a, b) = √ab, A(a, b) = (a + b)/2, and Q(a, b) = The following explicit formulas for NAG(a, b), NGA(a, b), NQA(a, b), and NAQ(a, b) are presented in [7]

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