Some operator Bellman type inequalities

In this paper, we employ the Mond--Pe\v{c}ari\'c method to establish some reverses of the operator Bellman inequality under certain conditions. In particular, we show \begin{equation*} \delta I_{\mathscr K}+\sum_{j=1}^n\omega_j\Phi_j\left((I_{\mathscr H}-A_j)^{p}\right)\ge \left(\sum_{j=1}^n\omega_j\Phi_j(I_{\mathscr H}-A_j)\right)^{p} \,, \end{equation*} where $A_j\,\,(1\leq j\leq n)$ are self-adjoint contraction operators with $0\leq mI_{\mathscr H}\le A_j \le MI_{\mathscr H}$, $\Phi_j$ are unital positive linear maps on ${\mathbb B}({\mathscr H})$, $\omega_j\in\mathbb R_+ \,\,(1\leq j\leq n)$, $0 < p < 1$ and $\delta=(1-p)\left(\frac{1}{p}\frac{(1-m)^p-(1-M)^p}{M-m}\right)^{\frac{p}{p-1}}+\frac{(1-M)(1-m)^p-(1-m)(1-M)^p}{M-m}$ . We also present some refinements of the operator Bellman inequality.