‎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.

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