Some operator Bellman type inequalities

Abstract

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 \begin{equation*} δI_{\mathscr K}+\sum_{j=1}^nω_jΦ_j\left((I_{\mathscr H}-A_j)^{p}\right)\ge \left(\sum_{j=1}^nω_jΦ_j(I_{\mathscr H}-A_j)\right)^{p} \,, \end{equation*} where $A_j(1\le j\le n)$ are self-adjoint contraction operators with $0\le m{I}_{\mathcal{H}}\le A_j\le M{I}_{\mathcal{H}}$, ${\Phi }_j$ are unital positive linear maps on $\mathbb{B}(\mathcal{H})$, ${\omega }_j\in {\mathbb{R}}_{+}(1\le j\le n)$, $0<p<1$ and $\delta =(1-p){\left(\frac{1}{p}\frac{(1-m{)}^p-(1-M{)}^p}{M-m}\right)}^{\frac{p}{p-1}}+\frac{(1-M)(1-m{)}^p-(1-m)(1-M{)}^p}{M-m}$ . We also present some refinements of the operator Bellman inequality.

arXiv comment15 pages, to appear in Indag. Math. (N.S.)