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\"older type inequalities. Among other results, it is shown that if A= (Aₜ) ₓ and B= (Bₜ) ₓ are continuous fields of positive invertible operators in a unital C^*-algebra A such that _Aₜ\, d (t) I ₀ and _Bₜ\, d (t) I ₀, and if f is an arbitrary operator mean with the representing function f, then align* (I ₀-_ (Aₜ f Bₜ) \, d (t) ) ᵖ (I ₀-_Aₜ\, d (t) ) ₅㵵 (I ₀-_Bₜ\, d (t) ) align* for all 0 p 1, which is an extension of the operator Bellman inequality.