  In the area of extremal graph theory, there exists a problem that
investigates the maximum induced density of a $k$-vertex graph $H$ in any
$n$-vertex graph $G$. This is known as the problem of \emph{inducibility} that
was first introduced by Pippenger and Golumbic in 1975. In this paper, we give
a new upper bound for the inducibility for a family of \emph{Double Loop
Graphs} of order $k$. The upper bound obtained for order $k=5$ is within a
factor of 0.964506 of the exact inducibility, and the upper bound obtained for
$k=6$ is within a factor of 3 of the best known lower bound.
