The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

Abstract

For $p\in ℝ$, the power mean $M_p(a,b)$ of order $p$, logarithmic mean $L(a,b)$, and arithmetic mean $A(a,b)$ of two positive real values $a$ and $b$ are defined by $M_p(a,b)=\left(\right(a^p+b^p)/2{)}^{1/p}$, for $p\ne 0$ and $M_p(a,b)=\sqrt{ab}$, for $p=0$, $L(a,b)=(b-a)/(\log b-\log a)$, for $a\ne b$ and $L(a,b)=a$, for $a=b$ and $A(a,b)=(a+b)/2$, respectively. In this paper, we answer the question: for $\alpha \in \left(\mathrm{0,1}\right)$, what are the greatest value $p$ and the least value $q$, such that the double inequality $M_p(a,b)\le \alpha A(a,b)+(1-\alpha )L(a,b)\le M_q(a,b)$ holds for all $a,b>0$.