On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in ${\mathbb{R}}^{2n}$

In this paper, we prove that for every dynamically convex compact star-shaped hypersurface $\mathrm{\Sigma }\subset {\mathbb{R}}^{2n}$, there exist at least $\lfloor \frac{n+1}{2}\rfloor$ geometrically distinct closed characteristics possessing irrational mean indices provided the number of geometrically distinct closed characteristics on $\mathrm{\Sigma }$ is finite, this improves Theorem 1.3 in \cite{LoZ} of Y. Long and C. Zhu by finding one more closed characteristic possessing irrational mean index when $n$ is odd. Moreover, there exist at least $\lfloor \frac{n+1}{2}\rfloor +1$ geometrically distinct closed characteristics such that the ratio of the mean indices of any two of them is a irrational number provided the number of geometrically distinct closed characteristics on $\mathrm{\Sigma }$ is finite, this improves Theorem 1.2 in \cite{HuO} of X. Hu and Y. Ou when $n$ is odd. In particular, these estimates are sharp for $n=3$.