We prove that for and all with if and only if and if and only if , and the parameter is the best possible in either case. Here, , , and and are the harmonic, logarithmic, and pth power means of a and b, respectively.
"Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means." J. Appl. Math. 2012 1 - 14, 2012. https://doi.org/10.1155/2012/471096