Optimal Lower Power Mean Bound for the Convex Combination of Harmonic and Logarithmic Means

Abstract

We find the least value $\lambda \in (0,1)$ and the greatest value $p=p\left(\alpha \right)$ such that $\alpha H(a,b)+(1-\alpha )L(a,b)>M_p(a,b)$ for $\alpha \in [\lambda ,1)$ and all $a,b>0$ with $a\ne b$, where $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ are the harmonic, logarithmic, and $p$-th power means of two positive numbers $a$ and $b$, respectively.