Sharp One-Parameter Mean Bounds for Yang Mean. (29th February 2016) - Qian, Wei-Mao; Chu, Yu-Ming; Zhang, Xiao-Hui
Abstract : We prove that the double inequalityJ α ( a, b ) < U ( a, b ) < J β ( a, b ) holds for alla, b > 0 witha ≠ b if and only ifα ≤ 2 / ( π - 2 ) = 0.8187 ⋯ andβ ≥ 3 / 2, whereU ( a, b ) = ( a - b ) / [ 2 arctan ⁡ ( ( a - b ) / 2 a b ) ], andJ p ( a, b ) = p ( a p + 1 - b p + 1 ) / [ ( p + 1 ) ( a p - b p ) ] ( p ≠ 0, - 1 ), J 0 ( a, b ) = ( a - b ) / ( log ⁡ a - log ⁡ b ), andJ - 1 ( a, b ) = a b ( log ⁡ a - log ⁡ b ) / ( a - b ) are the Yang andp th one-parameter means ofa andb, respectively.