  In this paper the Jessen's type inequality for normalized positive
$C_0$-semigroups is obtained. An adjoint of Jessen's type inequality has also
been derived for the corresponding adjoint-semigroup, which does not give the
analogous results but the behavior is still interesting. Moreover, it is
followed by some results regarding positive definiteness and exponential
convexity of complex structures involving operators from a semigroup.
