Bellman inequality for Hilbert space operators

We establish some operator versions of Bellman’s inequality. In particular, we prove that if Φ:B(H)→B(K)Φ:B(H)→B(K) is a unital positive linear map, A,B∈B(H)A,B∈B(H) are contractions, p>1p>1 and 0⩽λ⩽10⩽λ⩽1, then(Φ(IH-A∇λB))1/p⩾Φ((IH-A)1/p∇λ(IH-B)1/p).Φ(IH-A∇λB)1/p⩾Φ(IH-A)1/p∇λ(IH-B)1/p.