Some results of reverses Young's inequalities

Abstract

In this paper, we present some refinements of reverse Young's inequalities. Among other results, a refinement of reverse operator Young inequalities says
$$
A\nabla_{v}B+2\lambda(A\nabla B-A\sharp B) \leq \frac{m\nabla_{\lambda}M}{m\sharp_{\lambda}M}A\sharp_{v}B,
$$
where $0<mI\leq A,B\leq MI$, $\lambda=\min\{v,1-v\}$ and $v\in[0,1]$, extending a key result in [J. Math. Anal. Appl. {465} (2018) 267-280] and [Linear Multilinear Algebra {67} (2019) 1567-1578]. Furthermore, we give a reverse of Young's inequalities due to [Math. Slovaca 70 (2020), 453-466]. Moreover, we give a generalization of reverse Young-type inequality, and we also show a new Young-type inequality which is either better or not uniformly better than the main results in [Rocky Mountain J. Math. 46 (2016), 1089-1105]. As applications of these results, we obtain some inequalities for operators, Hilbert-Schmidt norms, unitarily invariant norms and determinants.