Some operator Bellman type inequalities

الأصل · EN

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 equation* δI K+∑ⱼ₌₁ⁿωⱼΦⱼ((I H-Aⱼ)ᵖ)≥ (∑ⱼ₌₁ⁿωⱼΦⱼ(I H-Aⱼ))ᵖ, equation* where Aⱼ(1≤ j≤ n) are self-adjoint contraction operators with 0≤ mI H≤ Aⱼ ≤ MI H, Φⱼ are unital positive linear maps on B(H), ωⱼ+ (1≤ j≤ n), 0 < p < 1 and δ=(1-p)(1/p(1-m)ᵖ-(1-M)ᵖ/M-m)ᵖ/ᵖ⁻¹+(1-M)(1-m)ᵖ-(1-m)(1-M)ᵖ/M-m. We also present some refinements of the operator Bellman inequality.