The Davis--Choi--Jensen inequality states that if f is an operator convex function on an interval J , then for every self-adjoint operator A acting on a Hilbert space mathcal H with spectra in J and each unital positive linear map Phi on mathcal B( mathcal H) , begin{eqnarray} label{dav} f( Phi(A)) leq Phi(f(A)) end{eqnarray} holds. In particular, begin{eqnarray} label{han} f( sum_{i=1}^n A_i^*X_iA_i) leq sum_{i=1}^n A_i^*f(X_i)A_i end{eqnarray} for every n -tuple (X_1, cdots,X_n) of elements of mathcal B( mathcal H) with spectra in J and every n -tuple (A_1,cdots,A_n) of operators in mathcal B( mathcal H) with sum_{i=1}^n A_i^*A_i= I . Also inequality ( ref{han}) is true if 0 in J , f(0) leq 0 and sum_{i=1}^n A_i^*A_i leq I . It is known that ( ref{dav}) is equivalent to the operator convexity of f . In this talk, we discuss some results concerning with Jensen s inequality, some equivalent conditions to the operator convexity, inequalities involving eigenvalues and present some refinements of the Choi--Davis--Jensen inequality for strictly positive maps.

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