Improved Jensen-type inequalities via linear interpolation and   applications
  In this paper we develop a general method for improving Jensen-type inequalities for convex and, even more generally, for piecewise convex functions. Our main result relies on the linear interpolation of a convex function. As a consequence, we obtain improvements of some recently established Young-type inequalities. Finally, our method is also applied to matrix case. In such a way we obtain improvements of some important matrix inequalities known from the literature. 
Daeshik Choi, Mario Krni\'c, and Josip Pecari\'c
math.CA