IMPROVED BELLMAN AND ACZEL INEQUALITIES FOR OPERATORS

We derive several more accurate operator Bellman and Aczel inequalities. Among other inequalities, it is shown that if Phi : B(H) -> B(H) is a unital positive linear map, A, B is an element of B(H) are two contraction operators, and we take 0 <= lambda, v <= 1 and p > 1, then Phi((I - A)(1-p)del(lambda) (I - B)(1/p)) <= (Phi(I - A del B-lambda))(1/p), which nicely improves the operator Bellman inequality.