Two optimal inequalities for Neuman-Sándor means

doi:10.3969/j.issn.1000-5641.2018.04.003

Abstract

Abstract: This paper deals with the inequalities involving Neuman-Sándor means using methods of real analysis. The convex combinations of the second contra-harmonic mean D(a, b) and the harmonic root-square mean H(a, b) (or harmonic mean H(a,b)) for the Neuman-Sándor mean M(a, b) are discussed. We find the maximum values λ₁, λ₂ ∈ (0, 1) and the minimum values μ₁, μ₂ ∈ (0, 1) such that the two-sided inequalities
λ₁D(a, b) + (1-λ₁)H(a, b) < M(a, b) < μ₁D(a, b) + (1-μ₁)H(a, b),
λ₂D(a, b) + (1-λ₂)H(a,b) < M(a, b) < μ₂D(a, b) + (1-μ₂)H(a,b)
hold for all a, b > 0 with a≠b.

Key words: Neuman-Sándor mean, contra-harmonic mean, root-square mean, harmonic mean, inequalities

CLC Number:

O178

YANG Yue-ying, MA Ping. Two optimal inequalities for Neuman-Sándor means[J]. Journal of East China Normal University(Natural Sc, 2018, 2018(4): 23-31.

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