  In this paper, we firstly generalize some theories developed by I. Ekeland
and H. Hofer in [EkH] for closed characteristics on compact convex
hypersurfaces in ${\bf R}^{2n}$ to star-shaped hypersurfaces. As applications,
we use Ekeland-Hofer theory and index iteration theory to prove that if a
compact star-shaped hypersuface in ${\bf R}^4$ satisfying some suitable
pinching condition carries exactly two geometrically distinct closed
characteristics, then both of them must be elliptic. We also conclude that the
theory developed by Y. Long and C. Zhu in [LoZ] still holds for dynamically
convex star-shaped hypersurfaces, and combining it with the results in [WHL],
[LLW], [Wan3], we obtain that there exist at least $n$ closed characteristics
on every dynamically convex star-shaped hypersurface in ${\bf R}^{2n}$ for
$n=3, 4$.
