  We establish some operator versions of Bellman's inequality. In particular,
we prove that if $\Phi: \mathbb{B}(\mathscr{H}) \to \mathbb{B}(\mathscr{K})$ is
a unital positive linear map, $A,B \in \mathbb{B}(\mathscr{H})$ are
contractions, $p>1$ and $0 \leq \lambda \leq 1$, 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*}
