Extensions of the operator Bellman and operator Holder type inequalities

In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator H\"{o}lder type inequalities. Among other results, it is shown that if A=(Aₜ)t∈ Ω and B=(Bₜ)t∈ Ω are continuous fields of positive invertible operators in a unital C^*-algebra A such that ∫ΩAₜ\,dμ(t)≤ IA and ∫ΩBₜ\,dμ(t)≤ IA, and if ωf is an arbitrary operator mean with the representing function f, then {align*} (IA-∫Ω(A_t ω_f B_t)\,dμ(t))^p ≥(IA-∫ΩA_t\,dμ(t)) ωf^p(IA-∫ΩB_t\,dμ(t)) {align*} for all 0 < p ≤ 1, which is an extension of the operator Bellman inequality.