In mathematics, the Neuman–Sándor mean of two positive, unequal numbers and is defined as
This mean satisfies the inequality , where is the unweighted arithmetic mean and is the second Seiffert mean
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.Tiehong Zhao, Yuming Chu and Baoyu Liu. Some Best Possible Inequalities Concerning Certain Bivariate Means. October 15, 2012. Several authors obtained sharp and optimal bounds for the Neuman–Sándor mean.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/10.1080/23311835.2014.995951Hui 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-08-20Huang, 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/s13660-015-0955-2Chu, 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-2013-10Tie-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 Neuman and others utilized this mean to study other bivariate means and inequalities.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, 637–643Y.-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), 567-577E. Neuman, A one-parameter family of bivariate mean, J. Math. Inequal. 7 (2013), No. 3, 399–412E. Neuman, Sharp inequalities involving Neuman–Sándor and logarithmic means, J. Math. Inequal. 7 (2013), No. 3, 413–419
See also
Mean Arithmetic mean Geometric mean Stolarsky mean Identric mean Means in Mathematical AnalysisGheorghe Toader and Iulia Costin. 2017. Means in Mathematical Analysis: Bivariate Means. 1st Edition. Academic Press. eBook , Paperback . https://www.elsevier.com/books/means-in-mathematical-analysis/toader/978-0-12-811080-5
References