  Let $\Phi$ 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 \[ |\Phi(f(A))\Phi(A)\Phi(g(A))|\leq U^*\Phi(f(A)Ag(A))U \] and
\[\left|\Phi\left(f(A)\right)^{-r}\Phi(A)^r\Phi\left(g(A)\right)^{-r}\right|\leq
V^*\Phi\left(f(A)^{-r}A^rg(A)^{-r}\right)V\] hold under some mild operator
convex conditions and some positive numbers $r$. Further, we show that if $f^2$
is operator concave, then $$ |\Phi(f(A))\Phi(A)|\leq \Phi(Af(A)).$$ 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.
