A Best Possible Double Inequality for Power Mean

Abstract

We answer the question: for any $p,q\in ℝ$ with $p\ne q$ and $p\ne -q$, what are the greatest value $\lambda =\lambda (p,q)$ and the least value $\mu =\mu (p,q)$, such that the double inequality holds for all $a,b>0$ with $a\ne b$? Where $M_p(a,b)$ is the $p$th power mean of two positive numbers $a$ and $b$.