  The Jensen inequality has been recognized as a powerful tool to deal with the
stability of time-delay systems. Recently, a new inequality that encompasses
the Jensen inequality was proposed for the stability analysis of systems with
finite delays. In this paper, we first present a generalized integral
inequality and its double integral extension. It is shown how these
inequalities can be applied to improve the stability result for linear
continuous-time systems with gamma-distributed delays. Then, for the
discrete-time counterpart we provide an extended Jensen summation inequality
with infinite sequences, which leads to less conservative stability conditions
for linear discrete-time systems with poisson-distributed delays. The
improvements obtained thanks to the introduced generalized inequalities are
demonstrated by examples.
