  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.
