Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means

Abstract

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∈(0,3345/80−11/16), 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 Mp(a,b)=((ap+bp)/2)1/p (p≠0) and M0(a,b)=ab are the harmonic, logarithmic, and pth power means of a and b, respectively.