  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 {equation*} \sum_{i=1}^{n} f(C_i) \leq
f(\sum_{i=1}^{n}C_i)-\delta_f\sum_{i=1}^{n}\widetilde{C}_i\leq
f(\sum_{i=1}^{n}C_i) {equation*} for all operators $C_i$ such that $0 \leq
C_i\leq M \leq \sum_{i=1}^{n} C_i $ \ $(i=1,...,n)$ for some scalar $M\geq0$,
where $ \widetilde{C_i} = 1/2 - |\frac{C_i}{M}- 1/2 |$ and $\delta_f =
f(0)+f(M) - 2 f(\frac{M}{2})$.
