Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means. (25th March 2013) - Zhao, Tie-Hong; Chu, Yu-Ming; Jiang, Yun-Liang; Li, Yong-Min
Abstract : We prove that the double inequalities I α 1 ( a, b ) Q 1 - α 1 ( a, b ) < M ( a, b ) < I β 1 ( a, b ) Q 1 - β 1 ( a, b ), I α 2 ( a, b ) C 1 - α 2 ( a, b ) < M ( a, b ) < I β 2 ( a, b ) C 1 - β 2 ( a, b ) hold for all a, b > 0 with a ≠ b if and only if α 1 ≥ 1 / 2, β 1 ≤ log [ 2 log ( 1 + 2 ) ] / ( 1 - log 2 ), α 2 ≥ 5 / 7, and β 2 ≤ log [ 2 log ( 1 + 2 ) ], where I ( a, b ), M ( a, b ), Q ( a, b ), and C ( a, b ) are the identric, Neuman-Sándor, quadratic, and contraharmonic means of a and b, respectively.