Some operator Bellman type inequalities

In this paper, we employ the Mond–Pečarić method to establish some reverses of the operator Bellman inequality under certain conditions. In particular, we show δIℋ+∑j=1nωjΦj((Iℋ−Aj)p)≥(∑j=1nωjΦj(Iℋ−Aj))p, where Aj(1≤j≤n) are self-adjoint contraction operators with 0≤mIℋ≤Aj≤MIℋ0≤mIℋ≤Aj≤MIℋ, ΦjΦj are unital positive linear maps on B(ℋ)B(ℋ), ωj∈R+(1≤j≤n) such that ∑j=1nωj=1, δ=(1−p)(1p(1−m)p−(1−M)pM−m)pp−1+(1−M)(1−m)p−(1−m)(1−M)pM−m and 0