Some inequalities involving weighted power mean
Abstract
In this paper, we firstly show some inequalities on weighted power mean. When $a,b>0$, $p\geq 1$ and $0 < v \leq \tau < 1$, we have
\[\frac{v}{\tau } \le \frac{{a{\sharp _{p,v}}b - a{\sharp _v}b}}{{a{\sharp _{p,\tau }}b - a{\sharp _\tau }b}} \le \frac{{1 - v}}{{1 - \tau }}\]
and
\[\frac{v}{\tau } \le \frac{{a{\sharp _{p,v}}b - a{!_v}b}}{{a{\sharp _{p,\tau }}b - a{!_\tau }b}} \le \frac{{1 - v}}{{1 - \tau }}.\]
Furtherly, we obtain several the corresponding inequalities involving the $m$ power form of weighted power mean in the same form as above for $m\in {\mathbb{N}}^+$ or $p\geq m>0$ ~,~ $m\leq p<0$. As applications, we give some inequalities about matrices and determinants respectively.
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Subscribers OnlyDOI: https://doi.org/10.2478/tmmp-2024-0019