Improved Bellman and Aczel inequalities for operators

We derive several more accurate operator Bellman and Aczel inequalities. Among other inequalities, it is shown that if $\Phi :\mathbb {B}( \mathscr {H} )\to \mathbb {B}( \mathscr {H})$ is a unital positive linear map, $A,B\in \mathbb {B}( \mathscr {H})$ are two contraction operators, and we take $0\le \lambda ,v\le 1$ and $p>1$, then $$ \Phi ( {{( I-A )}^{\frac {1}{p}}}{{\nabla }_{\lambda }}{{( I-B )}^{\frac {1}{p}}} ) \le {{( \Phi ( I-A{{\nabla }_{\lambda }}B ) )}^{\frac {1}{p}},} $$ which nicely improves the operator Bellman inequality.