Sharp Power Mean Bounds for the One-Parameter Harmonic Mean. (30th April 2015) - Chu, Yu-Ming; Wu, Li-Min; Song, Ying-Qing
Abstract : We present the best possible parametersα = α ( r ) andβ = β ( r ) such that the double inequalityM α ( a, b ) < H r ( a, b ) < M β ( a, b ) holds for allr ∈ ( 0, 1 / 2 ) anda, b > 0 witha ≠ b, whereM p ( a, b ) = [ ( a p + b p ) / 2 ] 1 / p ( p ≠ 0 ) andM 0 ( a, b ) = a b andH r ( a, b ) = 2 [ r a + ( 1 - r ) b ] [ r b + ( 1 - r ) a ] / ( a + b ) are the power and one-parameter harmonic means ofa andb, respectively.