In this paper we obtain several operator inequalities providing upper bounds for the Davis-Choi-Jensen's Difference $\Phi ((A)) - (\Phi(A))$ for any convex function $f:I\rightarrow \mathbb{R}$, any selfadjoint operator $A$ in $H$ with the spectrum $\limfunc{Sp}\left( A\right) \subset I$ and any linear, positive and normalized map $\Phi :\mathcal{B}\left(H\right) \rightarrow \mathcal{B}\left(K\right)$, where $H$ and $K$ are Hilbert spaces. Some examples of convex and operator convex functions are also provided.