Operator inequalities of Jensen type

Abstract

We present some generalized Jensen type operator inequalities<br><br>involving sequences of self-adjoint operators. Among other things,<br><br>we prove that if f:[0, infty) to mathbb{R} is a continuous<br><br>convex function with f(0) leq 0 , then<br><br> begin{equation*}<br><br> 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)<br><br> end{equation*}<br><br> 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<br><br> 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(<br><br> frac{M}{2} right).