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Neuman–Sándor mean

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In mathematics of special functions, the Neuman–Sándor mean M, of two positive and unequal numbers a and b, is defined as:

M ( a , b ) = a − b 2 arsinh ⁡ ( a − b a + b ) {\displaystyle M(a,b)={\frac {a-b}{2\operatorname {arsinh} \left({\frac {a-b}{a+b}}\right)}}}

This mean interpolates the inequality of the unweighted arithmetic mean A = (a + b)/2) and of the second Seiffert mean T defined as:

T ( a , b ) = a − b 2 arctan ⁡ ( a − b a + b ) , {\displaystyle T(a,b)={\frac {a-b}{2\arctan \left({\frac {a-b}{a+b}}\right)}},}

so that A < M < T.

The M(a,b) mean, introduced by Edward Neuman and József Sándor, has recently been the subject of intensive research and many remarkable inequalities for this mean can be found in the literature. Several authors obtained sharp and optimal bounds for the Neuman–Sándor mean. Neuman and others utilized this mean to study other bivariate means and inequalities.

See also

References

  1. E. Neuman & J. Sándor. On the Schwab–Borchardt mean, Math Pannon. 14(2) (2003), 253–266. http://www.kurims.kyoto-u.ac.jp/EMIS/journals/MP/index_elemei/mp14-2/mp14-2-253-266.pdf
  2. Tiehong Zhao, Yuming Chu and Baoyu Liu. Some Best Possible Inequalities Concerning Certain Bivariate Means. October 15, 2012. arXiv:1210.4219
  3. Wei-Dong Jiang & Feng Qi. Sharp bounds for the Neuman-Sándor mean in terms of the power and contraharmonic means. January 9, 2015. https://www.cogentoa.com/article/10.1080/23311835.2014.995951
  4. Hui Sun, Tiehong Zhao, Yuming Chu and Baoyu Liu. A note on the Neuman-Sándor mean. J. of Math. Inequal. dx.doi.org/10.7153/jmi-08-20
  5. Huang, HY., Wang, N. & Long, BY. Optimal bounds for Neuman–Sándor mean in terms of the geometric convex combination of two Seiffert means. J Inequal Appl (2016) 2016: 14. https://doi.org/10.1186/s13660-015-0955-2
  6. Chu, YM., Long, BY., Gong, WM. et al. Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means. J Inequal Appl (2013) 2013: 10. https://doi.org/10.1186/1029-242X-2013-10
  7. Tie-Hong Zhao, Yu-Ming Chu, and Bao-Yu Liu, “Optimal Bounds for Neuman-Sándor Mean in Terms of the Convex Combinations of Harmonic, Geometric, Quadratic, and Contraharmonic Means,” Abstract and Applied Analysis, vol. 2012, Article ID 302635, 9 pages, 2012. doi:10.1155/2012/302635
  8. E. Neuman, Inequalities for weighted sums of powers and their applications, Math. Inequal. Appl. 15 (2012), No. 4, 995–1005.
  9. E. Neuman, A note on a certain bivariate mean, J. Math. Inequal. 6 (2012), No. 4, 637–643
  10. Y.-M. Li, B.-Y. Long and Y.-M. Chu. Sharp bounds for the Neuman-Sándor mean in terms of generalized logarithmic mean. J. Math. Inequal. 6, 4(2012), 567-577
  11. E. Neuman, A one-parameter family of bivariate mean, J. Math. Inequal. 7 (2013), No. 3, 399–412
  12. E. Neuman, Sharp inequalities involving Neuman–Sándor and logarithmic means, J. Math. Inequal. 7 (2013), No. 3, 413–419
  13. Gheorghe Toader and Iulia Costin. 2017. Means in Mathematical Analysis: Bivariate Means. 1st Edition. Academic Press. eBook ISBN 9780128110812, Paperback ISBN 9780128110805. https://www.elsevier.com/books/means-in-mathematical-analysis/toader/978-0-12-811080-5
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