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

Abstract

We prove that $\alpha H(a,b)+(1-\alpha )L(a,b)>M_{(1-4\alpha )/3}(a,b)$ for $\alpha \in (0,1)$ and all $a,b>0$ with $a\ne b$ if and only if $\alpha \in [1/4,1)$ and if and only if $\alpha \in (0,3\sqrt{345}/80-11/16)$, and the parameter $(1-4\alpha )/3$ is the best possible in either case. Here, $H(a,b)=2ab/(a+b)$, $L(a,b)=(a-b)/(\text{log}\mathrm{\hspace{0.17em}a}-\text{log}\mathrm{\hspace{0.17em}b})$, and $M_p(a,b)=((a^p+b^p)/2{)}^{1/p}(p\ne 0)$ and $M_0(a,b)=\sqrt{ab}$ are the harmonic, logarithmic, and pth power means of a and b, respectively.