  In this paper, we first generalize the common index jump theorem of Long-Zhu
in 2002 and Duan-Long-Wang in 2016 to the case where the mean indices of
symplectic paths are not required to be all positive. As applications, we study
closed characteristics on compact star-shaped hypersurfaces in ${\bf R}^{2n}$,
when both positive and negative mean indices may appear simultaneously.
Specially we establish the existence of at least $n$ geometrically distinct
closed characteristics on every compact non-degenerate perfect star-shaped
hypersurface $\Sigma$ in ${\bf R}^{2n}$ provided that every prime closed
characteristic possesses nonzero mean index. Furthermore, in the case of ${\bf
R}^6$ we remove the nonzero mean index condition by showing that the existence
of only finitely many geometrically distinct closed characteristics implies
that each of them must possess nonzero mean index. We also generalize the above
results about closed characteristics on non-degenerate star-shaped
hypersurfaces to closed Reeb orbits of non-degenerate contact forms on a broad
class of prequantization bundles.
