Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means

Abstract

We answer the question: for $\alpha \in \left(0,1\right)$, what are the greatest value $p$ and the least value $q$ such that the double inequality $M_p\left(a,b\right)<P^{\alpha }\left(a,b\right)G^{1-\alpha }\left(a,b\right)<M_q\left(a,b\right)$ holds for all $a,b>0$ with $a\ne b$. Here, $M_p\left(a,b\right)$, $P\left(a,b\right)$, and $G\left(a,b\right)$ denote the power of order $p$, Seiffert, and geometric means of two positive numbers $a$ and $b$, respectively.