A Sharp Double Inequality between Seiffert, Arithmetic, and Geometric Means. (2nd September 2012)

• Record Type: Journal Article
• Authors: Gong, Wei-Ming; Song, Ying-Qing; Wang, Miao-Kun; Chu, Yu-Ming
• Other Names: Diblík Josef Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2012(2012)
• Publication Date: 2012-09-02
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2012/684834
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : For fixed s ≥ 1 and any t 1, t 2 ∈ ( 0, 1 / 2 ) we prove that the double inequality G s ( t 1 a + ( 1 - t 1 ) b, t 1 b + ( 1 - t 1 ) a ) A 1 - s ( a, b ) < P ( a, b ) < G s ( t 2 a + ( 1 - t 2 ) b, t 2 b + ( 1 - t 2 ) a ) A 1 - s ( a, b ) holds for all a, b > 0 with a ≠ b if and only if t 1 ≤ ( 1 - 1 - ( 2 / π ) 2 / s ) / 2 and t 2 ≥ ( 1 - 1 / 3 s ) / 2 . Here, P ( a, b ), A ( a, b ) and G ( a, b ) denote the Seiffert, arithmetic, and geometric means of two positive numbers a and b, respectively.