Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A best-possible double inequality between Seiffert and harmonic means | Chu et al. Journal of Inequalities and Applications 2011 2011 94 http content 2011 1 94 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access A best-possible double inequality between Seiffert and harmonic means Yu-Ming Chu1 Miao-Kun Wang1 and Zi-Kui Wang2 Correspondence chuyuming2005@ department of Mathematics Huzhou Teachers College Huzhou 313000 China Full list of author information is available at the end of the article Abstract In this paper we establish a new double inequality between the Seiffert and harmonic means. The achieved results is inspired by the papers of Sándor Arch. Math. 76 34-40 2001 and Hasto Math. Inequal. Appl. 7 47-53 2004 and the methods from Wang et al. J. Math. Inequal. 4 581-586 2010 . The inequalities we obtained improve the existing corresponding results and in some sense are optimal. 2010 Mathematics Subject Classification 26E60. Keywords harmonic mean Seifert mean inequality 1 Introduction For a b 0 with a b the Seifert mean P a b was introduced by Seiffert 1 as follows P a b a b . 4 arctan y a b n Recently the bivariate mean values have been the subject of intensive research. In particular many remarkable inequalities for the Seiffert mean can be found in the literature 1-9 . Let H a b 2ab a b G a b Vab L a b a - b log a - log b I a b 1 e bb aa 1 b-a A a b a b 2 C a b a2 b2 a b and Mp a b ap bp 2 1 p p 0 and M0 a b Vab be the harmonic geometric logarithmic identric arithmetic contraharmonic and p-th power means of two different positive numbers a and b respectively. Then it is well known that min a b H a b M 1 a b G a b M0 a b L a b I a b A a b M1 a b C a b max a b . For all a b 0 with a b Seiffert 1 established that L a b P a b I a b Jagers 4 proved that M1 2 a b P a b M2 3 a b and M2 3 a b is the bestpossible upper power mean bound for the Seiffert mean P a b Seiffert 7 established that P a b A a b G a b L a b and P a b 2 A a b n Sándor 6
