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Báo cáo hóa học: " Two sharp double inequalities for Seiffert mean"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Two sharp double inequalities for Seiffert mean | Chu et al. Journal of Inequalities and Applications 2011 2011 44 Ú Journal of Inequalities and Applications http www.journalofinequalitiesandapplications.eom content 2011 1 44 a SpringeiOpen Journal RESEARCH Open Access Two sharp double inequalities for Seiffert mean Yu-Ming Chu1 Miao-Kun Wang1 and Wei-Ming Gong2 Correspondence chuyuming2005@yahoo.com.cn department of Mathematics Huzhou Teachers College Huzhou 313000 People s Republic of China Full list of author information is available at the end of the article Abstract In this paper we establish two new inequalities between the root-square arithmetic and Seiffert means. The achieved results are inspired by the paper of Seiffert Die Wurzel 29 221-222 1995 and the methods from Chu et al. J. Math. Inequal. 4 581-586 2010 . The inequalities we obtained improve the existing corresponding results and in some sense are optimal. Mathematics Subject Classification 2010 26E60. Keywords Root-square mean arithmetic mean Seiffert mean 1 Introduction For a b 0 with a b the root-square mean S a b and Seiffert mean T a b are defined by S a b -ị L 1.1 and M. a - b T a b - b. 1.2 2 arctan a J respectively. In the recent past both mean values have been the subject of intensive research. In particular many remarkable inequalities for S and T can be found in the literature 1-11 . Let A a b - a b 2 G a b - Tab and H a b 2ab a b be the classical arithmetic geometric and harmonic means of two positive numbers a and b respectively. In 1 Seiffert proved that A a b T a b S a b for all a b 0 with a b. Taneja 5 presented that G a b 2H a b 1 S a b 1 A a b 1 H a b 1 S a b 1 G a b 1 2 7 7 7 7 H a b -S a b A a b S a b - G a b H a b for all a b 0 with a b. Springer 2011 Chu et al licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution Jcense http creativecommons.org licenses by 2.0 which permits unrestricted use distribution and reproduction in any medium provided the original work is .