Optimal Bounds for Neuman Mean Using Arithmetic and Centroidal Means. (2nd February 2016) - Song, Ying-Qing; Qian, Wei-Mao; Chu, Yu-Ming
Abstract : We present the best possible parameters α 1, α 2, β 1, β 2 ∈ R and α 3, β 3 ∈ ( 1 / 2, 1 ) such that the double inequalities α 1 A ( a, b ) + ( 1 - α 1 ) C ( a, b ) < N Q A ( a, b ) < β 1 A ( a, b ) + ( 1 - β 1 ) C ( a, b ), A α 2 ( a, b ) C 1 - α 2 ( a, b ) < N Q A ( a, b ) < A β 2 ( a, b ) C 1 - β 2 ( a, b ), and C [ α 3 a + ( 1 - α 3 ) b, α 3 b + ( 1 - α 3 ) a ] < N Q A ( a, b ) < C [ β 3 a + ( 1 - β 3 ) b, β 3 b + ( 1 - β 3 ) a ] hold for all a, b > 0 with a ≠ b and give several sharp inequalities involving the hyperbolic and inverse hyperbolic functions. Here, N ( a, b ), A ( a, b ), Q ( a, b ), and C ( a, b ) are, respectively, the Neuman, arithmetic, quadratic, and centroidal means of a and b, and N Q A ( a, b ) = N [ Q ( a, b ), A ( a, b ) ] .