A Sharp Double Inequality between Seiffert, Arithmetic, and Geometric Means. (2nd September 2012) - Gong, Wei-Ming; Song, Ying-Qing; Wang, Miao-Kun; Chu, Yu-Ming
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.