The Optimal Geometric Combination Bounds for Neuman Means of Harmonic, Arithmetic and Contra-harmonic*

摘要

In the paper, we find the greatest values λ_1,λ_2,λ_3,λ_4 and the least values μ_1, μ_2, μ_3, μ_4 such that the double inequalities A~(λ_1)(a,b)H~(1-λ_1)(a,b) < N_(AG)(a,b) < A~(μ~1) (a, b)H~(1-μ~1) (a, b), C~(λ_2)(a,b)A~(1-λ_2)(a,b) < N_(QA)(a,b) < C~(μ~2) (a,b)A~(1-μ~2) (a,b), A~(λ_3)(a, b)H~(1-λ_3)(a, b) < N_(GA)(a,b) < A~(μ~3) (a, b)H~(1-μ~3)(a, b), C~(λ_4) (a, b)A~(1-λ_4) (a, b) < N_(AQ)(a,b) < C~(μ~4) (a,b)A~(1-μ~4) (a, b) hold for all a,b> 0 with a ≠ b. Here H(a, b), G(a, b), A(a, b), Q(a, b), C(a, b) respectively denote the harmonic, geometric, arithmetic, quadratic and contra-harmonic of a and b, and N_(AG)(a, b), N_(GA)(a, b), N_(QA)(a, b) and N_(AQ)(a, b) are four Neuman means derived from the Schwab-Borchardt mean.