Some extensions of the operator entropy type inequalities

In this paper, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond-Pe\v{c}ari\'c method. In particular, we present {\tiny \begin{align*} f&\left[\int_T(A_s\natural_{p+1}B_s)d\mu(s)+t_0\left(I_{\mathscr H}-\int_TA_s\natural_pB_sd\mu(s)\right)\right]-\gamma_ff(t_0)\left(I_{\mathscr H}-\int_TA_s\natural_pB_sd\mu(s)\right)\nonumber\\ &\le \gamma_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 $T$, $0<m A_s \leq B_s \leq M A_s\,\,(s\in T)$ for some positive real numbers $m, M$ such that $m<1<M$, $\int_TA_s=\int_TB_s=I_{\mathscr H}$, $f: (0,\infty) \to [0,\infty)$ be operator concave, $\gamma_f=\max\left\{\frac{f(t)}{\mu_f t+\nu_f}: m\leq t\leq M,\mu_f=\frac{f(M)-f(m)}{M-m}, \nu_f=\frac{Mf(m)-mf(M)}{M-m}\right\}$, $t_0\in[m,M]$, $p\in[0,1]$, and $$\widetilde{S}_p^f(\mathbf{A}|\mathbf{B})=\int_TA_s^{\frac{1}{2}}\left(A_s^{-\frac{1}{2}}B_sA_s^{-\frac{1}{2}}\right)^p f\left(A_s^{-\frac{1}{2}}B_sA_s^{-\frac{1}{2}}\right)A_s^{\frac{1}{2}}d\mu(s)\,.$$