Bellman inequality for Hilbert space operators

We establish some operator versions of Bellman’s inequality. In particular, we prove that if $\mathrm{\Phi }:\mathbb{B}(\mathcal{H})\to \mathbb{B}(\mathcal{K})$ is a unital positive linear map, $A,B\in \mathbb{B}(\mathcal{H})$ are contractions, $p>1$ and $0\le \lambda \le 1$, then {eqnarray*} \big(\Phi(I_\mathscr{H}-A\nabla_{\lambda}B)\big)^{1/p}\ge\Phi\big((I_\mathscr{H}-A)^{1/p}\nabla_{\lambda}(I_\mathscr{H}-B)^{1/p}\big). {eqnarray*}