K-stability of Fano manifolds with not small alpha invariants

We show that any $n$-dimensional Fano manifold $X$ with $\alpha (X)=n\mathrm{/}(n+1)$ and $n\ge 2$ is K-stable, where $\alpha (X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits K"ahler-Einstein metrics and the holomorphic automorphism group of $X$ is finite.