Journal of East China Normal University(Natural Science) >

2018 , Vol. 2018 >Issue 4: 23 - 31

DOI: https://doi.org/10.3969/j.issn.1000-5641.2018.04.003

Two optimal inequalities for Neuman-Sándor means

  • YANG Yue-ying ,
  • MA Ping
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  • Mechanic Electronic and Automobile Engineering College, Huzhou Vocational & Technical College, Huzhou Zhejiang 313000, China

Received date: 2017-03-27

  Online published: 2018-07-19

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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

Cite this article

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

References

[1] NEUMAN E, SÁNDOR J. On the Schwab-Borchardt mean[J]. Mathematica Pannonica, 2003, 14(2):253-266.
[2] LI Y M, LONG B Y, CHU Y M. Sharp bounds for the Neuman-Sándor mean in terms of generalized logarithmic mean[J]. Journal of Mathematical Inequalities, 2012, 6(4):567-577.
[3] CHU Y M, LONG B Y, GONG W M, et al. Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means[J]. Journal of Inequalities and Applications, 2013, 2013:10.
[4] CHU Y M, LONG B Y. Bounds of the Neuman-Sándor mean using power and identric means[J]. Abstract and Applied Analysis, 2013, 6pages.
[5] ZHAO T H, CHU Y M, JIANG Y L, et al. Best possible bounds for Neuman-Sándor mean by the identric, quadratic and contraharmonic means[J]. Abstract and Applied Analysis, 2013, 12pages.
[6] HE Z Y, QIAN W M, JIANG Y L, et al. Bounds for the combinations of Neuman-Sándor, arithmetic and second Seiffert means in terms of contra-harmonic mean[J]. Abstract and Applied Analysis, 2013, 5pages.
[7] NEUMAN E, SÁNDOR J. On the Schwab-Borchardt mean Ⅱ[J]. Mathematica Pannonica, 2006, 17(1):49-59.
[8] NEUMAN E. A note on a certain bivariate mean[J]. Journal of Mathematical Inequalities, 2012, 6(4):637-643.
[9] ZHAO T H, CHU Y M, LIU B Y. Optimal bounds for Neuman-Sándor mean in terms of the convex combinations of harmonic, geometric, quadratic, and contraharmonic means[J]. Abstract and Applied Analysis, 2012, 9pages.
[10] QIAN W M, CHU Y M. On certain inequalities for Neuman-Sándor mean[J]. Abstract and Applied Analysis, 2013, 6pages.
[11] QIAN W M, SONG Y Q, ZHANG X H, et al. Sharp bounds for Toader mean in terms of arithmetic and second contra-harmonic means[J]. Journal of Function Spaces, 2015, 5pages.
[12] 孟祥菊, 刘红, 高红亚. 第二类,Seiffert,平均的最优凸组合界[J]. 宁夏大学学报(自然科学版), 2012, 33(1):14-16.
[13] 孟祥菊, 王淑燕, 田淑环. 关于第二类,Seiffert,平均的最佳双边不等式[J]. 数学的实践与认识, 2015, 45(18):299-302.
[14] YANG Y Y, SHEN L C, QIAN W M. The optimal convex combination bounds of second contra-harmonic and geometric mean for the Seiffert means[J]. Pacific Journal of Applied Mathematics, 2016, 7(3):207-217.
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