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

Wei-Mao Qian,Yu-Ming Chu

doi:10.1186/1029-242x-2014-175

Wei-Mao Qian, Yu-Ming Chu

Open Access

https://doi.org/10.1186/1029-242x-2014-175

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Abstract

In this paper, we present sharp bounds for the two Neuman means S H A and S C A 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.

Highlights

Let a, b > with a = b, the Schwab-Borchardt mean SB(a, b) is defined by ⎧√ SB(a, b) = ⎨ ⎩b –a co√s– (a/b), a –b cosh– (a/b), a < b, a > b, ( . )√ where cos– (x) and cosh– (x) = log(x + x – ) are the inverse cosine and inverse hyperbolic cosine functions, respectively.√ where G(a, b) = ab, A(a, b) = (a + b)/ and Q(a, b) = (a + b )/ denote the classical geometric mean, arithmetic mean and quadratic mean of a and b, respectively
In [ ], the author proved that the double inequalities αQ(a, b) + ( – α)A(a, b) < M(a, b) < βQ(a, b) + ( – β)A(a, b) and λC(a, b) + ( – λ)A(a, b) < M(a, b) < μC(a, b) + ( – μ)A(a, b) hold for all a, b > with a = b if and √only if α ≤ [ √– log( + )]/[( – ) log( + )] =
Let v = (a – b)/(a + b) ∈ (, ), the following explicit formulas for SAH, SHA, SAC and SCA have been found by Neuman [ ]: tanh(p)

Summary

Introduction

1 Introduction Let a, b > with a = b, the Schwab-Borchardt mean SB(a, b) is defined by SB(a, b) In [ ], the author proved that the double inequalities αQ(a, b) + ( – α)A(a, b) < M(a, b) < βQ(a, b) + ( – β)A(a, b) and λC(a, b) + ( – λ)A(a, b) < M(a, b) < μC(a, b) + ( – μ)A(a, b) hold for all a, b > with a = b if and √only if α ≤ [ √– log( + )]/[( – ) log( + )] = We call the means SAH , SHA, SCA and SAC given in

Results

Conclusion