We present some generalized Jensen type operator inequalities involving sequences of self-adjoint operators. Among other things, we prove that if f:[0, infty) to mathbb{R} is a continuous convex function with f(0) leq 0 , then begin{equation*} sum_{i=1}^{n} f(C_i) leq f left( sum_{i=1}^{n}C_i right)- delta_fsum_{i=1}^{n} widetilde{C}_i leq f left( sum_{i=1}^{n}C_i right) end{equation*} for all operators C_i such that 0 leq C_i leq M leq sum_{i=1}^{n} C_i (i=1, ldots,n) for some scalar M geq0 , where widetilde{C_i} = frac{1}{2} - left|frac{C_i}{M}- frac{1}{2} right| and delta_f = f(0)+f(M) - 2 f left( frac{M}{2} right).

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