An Optimal Double Inequality between Power-Type Heron and Seiffert Means. (31st October 2010) - Chu, Yu-Ming; Wang, Miao-Kun; Qiu, Ye-Fang
Abstract : Fork ∈ [ 0, + ∞ ), the power-type Heron meanH k ( a, b ) and the Seiffert meanT ( a, b ) of two positive real numbersa andb are defined byH k ( a, b ) = ( ( a k + ( a b ) k / 2 + b k ) / 3 ) 1 / k, k ≠ 0 ;H k ( a, b ) = a b, k = 0 andT ( a, b ) = ( a - b ) / 2 arctan ( ( a - b ) / ( a + b ) ), a ≠ b ;T ( a, b ) = a, a = b, respectively. In this paper, we find the greatest valuep and the least valueq such that the double inequalityH p ( a, b ) < T ( a, b ) < H q ( a, b ) holds for alla, b > 0 witha ≠ b .