Some extensions of the operator entropy type inequalities

Abstract

In this paper, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond-Pečarić method. In particular, we present {\tiny \begin{align*} f&\left[\int_T(A_s\natural_{p+1}B_s)dμ(s)+t_0\left(I_{\mathscr H}-\int_TA_s\natural_pB_sdμ(s)\right)\right]-γ_ff(t_0)\left(I_{\mathscr H}-\int_TA_s\natural_pB_sdμ(s)\right)\nonumber\\ &\le γ_f\widetilde{S}_p^f(\mathbf{A}|\mathbf{B})\,, \end{align*}} where $T$ is a locally compact Hausdorff space and $\mu$ is a Radon measure on $T$, $0<m{A}_s\le B_s\le M{A}_s(s\in T)$ for some positive real numbers $m,M$ such that $m<1<M$, ${\int }_T{A}_s={\int }_T{B}_s=I_{\mathcal{H}}$, $f:(0,\infty )\to [0,\infty )$ be operator concave, ${\gamma }_f=\max \left\{\frac{f(t)}{{\mu }_{f}t+{\nu }_f}:m\le t\le M,{\mu }_f=\frac{f(M)-f(m)}{M-m},{\nu }_f=\frac{Mf(m)-mf(M)}{M-m}\right\}$, $t_0\in [m,M]$, $p\in [0,1]$, and $${\tilde{S}}_p^f(\mathbf{A}|\mathbf{B})={\int }_T{A}_s^{\frac{1}{2}}{\left(A_s^{-\frac{1}{2}}{B}_s{A}_s^{-\frac{1}{2}}\right)}^{p}f\left(A_s^{-\frac{1}{2}}{B}_s{A}_s^{-\frac{1}{2}}\right)A_s^{\frac{1}{2}}d\mu (s).$$