Two optimal inequalities for Neuman-Sándor means
2018, 2018 (4):
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
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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.