On reverses of the Golden-Thompson type inequalities

In this paper we present some reverses of the Golden-Thompson type inequalities: Let $H$ and $K$ be Hermitian matrices such that $ e^s e^H \preceq_{ols} e^K \preceq_{ols} e^t e^H$ for some scalars $s \leq t$, and $\alpha \in [0 , 1]$. Then for all $p>0$ and $k =1,2,\ldots, n$ \begin{align*} \label{} \lambda_k (e^{(1-\alpha)H + \alpha K} ) \leq (\max \lbrace S(e^{sp}), S(e^{tp})\rbrace)^{\frac{1}{p}} \lambda_k (e^{pH} \sharp_\alpha e^{pK})^{\frac{1}{p}}, \end{align*} where $A\sharp_\alpha B = A^\frac{1}{2} \big ( A^{-\frac{1}{2}} B^\frac{1}{2} A^{-\frac{1}{2}} \big) ^\alpha A^\frac{1}{2}$ is $\alpha$-geometric mean, $S(t)$ is the so called Specht's ratio and $\preceq_{ols}$ is the so called Olson order. The same inequalities are also provided with other constants. The obtained inequalities improve some known results.