Inequalities between power means and convex combinations of the harmonic and logarithmic means

We prove that αh (a,b) + (1 - α) l (a,b) > m (1-4α)/3(a,b) for α ∈ (0,1) and all a,b > 0 with a ≠ b if and only if α ∈ [1/4,1) and αh (a,b) + (1 - α)l(a,b) < m (1-4α)/3(a,b) if and only if α ∈ (0,3√345/80 - 11/16),and the parameter (1 - 4α)/3 is the best possible in either case. here,h(a,b) = 2ab/(a + b),l (a,b) = (a - b)/(log a - log b),and m p (a,b) = ((a p + b p)/2) 1/p (p ≠ 0) and m 0 (a,b) = √ab are the harmonic,logarithmic,and pth power means of a and b,respectively. copyright © 2012 wei-mao qian and zhong-hua shen.