  We show that any $n$-dimensional Fano manifold $X$ with $\alpha(X)=n/(n+1)$
and $n\geq 2$ 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.
