Two elliptic closed geodesics on positively curved Finsler spheres

In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^n,F)$ with reversibility $\text{lm}$ and flag curvature $K$ satisfying ${\left(\frac{\text{lm}}{1+\text{lm}}\right)}^2<K\le 1$, either there exist infinitely many closed geodesics, or there exist at least two elliptic closed geodesics and each linearized Poincar'{e} map has at least one eigenvalue of the form $e^{\sqrt{-1}\mathrm{th}}$ with $\mathrm{th}$ being an irrational multiple of $\pi$.