Asymmetric Choi--Davis inequalities

Let $\mathrm{\Phi }$ be a unital positive linear map and let $A$ be a positive invertible operator. We prove that there exist partial isometries $U$ and $V$ such that $$\mathrm{\mid }\mathrm{\Phi }(f(A))\mathrm{\Phi }(A)\mathrm{\Phi }(g(A))\mathrm{\mid }\le U^{\ast }\mathrm{\Phi }(f(A)Ag(A))U$$ and $$\mid \mathrm{\Phi }{(f(A))}^{-r}\mathrm{\Phi }(A{)}^r\mathrm{\Phi }{(g(A))}^{-r}\mid \le V^{\ast }\mathrm{\Phi }(f(A{)}^{-r}{A}^{r}g(A{)}^{-r})V$$ hold under some mild operator convex conditions and some positive numbers $r$. Further, we show that if $f^2$ is operator concave, then $$\mathrm{\mid }\mathrm{\Phi }(f(A))\mathrm{\Phi }(A)\mathrm{\mid }\le \mathrm{\Phi }(Af(A))\mathrm{.}$$ In addition, we give some counterparts to the asymmetric Choi–Davis inequality and asymmetric Kadison inequality. Our results extend some inequalities due to Bourin–Ricard and Furuta.