Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means. (22nd February 2012) - Qian, Wei-Mao; Shen, Zhong-Hua
Abstract : We prove thatα H ( a, b ) + ( 1 − α ) L ( a, b ) > M ( 1 − 4 α ) / 3 ( a, b ) forα ∈ ( 0, 1 ) and alla, b > 0 witha ≠ b if and only ifα ∈ [ 1 / 4, 1 ) andα H ( a, b ) + ( 1 − α ) L ( a, b ) < M ( 1 − 4 α ) / 3 ( a, b ) if and only ifα ∈ ( 0, 3 345 / 80 − 11 / 16 ), and the parameter( 1 − 4 α ) / 3 is the best possible in either case. Here, H ( a, b ) = 2 a b / ( a + b ), L ( a, b ) = ( a − b ) / ( log a − log b ), andM p ( a, b ) = ( ( a p + b p ) / 2 ) 1 / p ( p ≠ 0 ) andM 0 ( a, b ) = a b are the harmonic, logarithmic, and p th power means of a and b, respectively.