Journal of East China Normal University(Natural Sc ›› 2018, Vol. 2018 ›› Issue (4): 23-31.doi: 10.3969/j.issn.1000-5641.2018.04.003

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Two optimal inequalities for Neuman-Sándor means

YANG Yue-ying, MA Ping   

  1. Mechanic Electronic and Automobile Engineering College, Huzhou Vocational & Technical College, Huzhou Zhejiang 313000, China
  • Received:2017-03-27 Online:2018-07-25 Published:2018-07-19

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 ab.

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

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