  In this paper we investigate the following questions. Let $\mu, \nu$ be two
regular Borel measures of finite total variation. When do we have a constant
$C$ satisfying $$\int f d\nu \le C \int f d\mu$$ whenever $f$ is a continuous
nonnegative positive definite function? How the admissible constants $C$ can be
characterized, and what is their optimal value? We first discuss the problem in
locally compact abelian groups. Then we make further specializations when the
Borel measures $\mu, \nu$ are both either purely atomic or absolutely
continuous with respect to a reference Haar measure. In addition, we prove a
duality conjecture posed in our former paper.
