Neuman–Sándor mean

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)={\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)={\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,^[1] has recently been the subject of intensive research and many remarkable inequalities for this mean can be found in the literature.^[2] Several authors obtained sharp and optimal bounds for the Neuman–Sándor mean.^[3]^[4]^[5]^[6]^[7] Neuman and others utilized this mean to study other bivariate means and inequalities.^[8]^[9]^[10]^[11]^[12]

References

^ 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
^ Tiehong Zhao, Yuming Chu and Baoyu Liu. Some Best Possible Inequalities Concerning Certain Bivariate Means. October 15, 2012. arXiv:1210.4219
^ 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
^ 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
^ 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
^ 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
^ 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
^ E. Neuman, Inequalities for weighted sums of powers and their applications, Math. Inequal. Appl. 15 (2012), No. 4, 995–1005.
^ E. Neuman, A note on a certain bivariate mean, J. Math. Inequal. 6 (2012), No. 4, 637–643
^ 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
^ E. Neuman, A one-parameter family of bivariate mean, J. Math. Inequal. 7 (2013), No. 3, 399–412
^ E. Neuman, Sharp inequalities involving Neuman–Sándor and logarithmic means, J. Math. Inequal. 7 (2013), No. 3, 413–419
^ 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

[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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See also

References