Two optimal inequalities for Neuman-Sándor means

YANG Yue-ying; MA Ping

doi:10.3969/j.issn.1000-5641.2018.04.003

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Jul. 2018

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YANG Yue-ying, MA Ping. Two optimal inequalities for Neuman-Sándor means[J]. Journal of East China Normal University (Natural Sciences), 2018, (4): 23-31. doi: 10.3969/j.issn.1000-5641.2018.04.003

Citation:

YANG Yue-ying, MA Ping. Two optimal inequalities for Neuman-Sándor means[J]. Journal of East China Normal University (Natural Sciences), 2018, (4): 23-31. doi: 10.3969/j.issn.1000-5641.2018.04.003

YANG Yue-ying, MA Ping. Two optimal inequalities for Neuman-Sándor means[J]. Journal of East China Normal University (Natural Sciences), 2018, (4): 23-31. doi: 10.3969/j.issn.1000-5641.2018.04.003

Citation:

YANG Yue-ying, MA Ping. Two optimal inequalities for Neuman-Sándor means[J]. Journal of East China Normal University (Natural Sciences), 2018, (4): 23-31. doi: 10.3969/j.issn.1000-5641.2018.04.003

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Two optimal inequalities for Neuman-Sándor means

doi: 10.3969/j.issn.1000-5641.2018.04.003

Mechanic Electronic and Automobile Engineering College, Huzhou Vocational & Technical College, Huzhou Zhejiang 313000, China

Received Date: 2017-03-27
Publish Date: 2018-07-25

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 $\overline{H}(a, b)$ (or harmonic mean $H(a, b)$) for the Neuman-Sándor mean $M(a, b)$ are discussed. We find the maximum values $\lambda_{1}, \lambda_{2}\in(0, 1)$ and the minimum values $\mu_{1}, \mu_{2}\in(0, 1)$ such that the two-sided inequalities $\lambda_{1}D(a,b)+(1-\lambda_{1})\overline{H}(a,b) <M(a,b)<\mu_{1}D(a,b)+(1-\mu_{1})\overline{H}(a,b), \\ \lambda_{2}D(a,b)+(1-\lambda_{2})H(a,b)<M(a,b)<\mu_{2}D(a,b)+(1-\mu_{2})H(a,b)$ hold for all $a, b>0$ with $a\neq b$.
- Neuman-Sándor mean,
- contra-harmonic mean,
- root-square mean,
- harmonic mean,
- inequalities

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References(14)

References

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