On K-stability of Fano weighted hypersurfaces

Let $X \subset \mathbb P(a_0,\ldots,a_n)$ be a quasi-smooth weighted Fano hypersurface of degree $d$ and index $I_X$ such that $a_i |d$ for all $i$, with $a_0 \le \ldots \le a_n$. If $I_X=1$, we show that, under a suitable condition, the $\alpha$-invariant of $X$ is greater than or equal to $\dim X/(\dim X+1)$ and $X$ is K-stable. This can be applied in particular to any $X$ as above such that $\dim X \le 3$. If $X$ is general and $I_X < \dim X$, then we show that $X$ is K-stable. We also give a sufficient condition for the finiteness of automorphism groups of quasi-smooth Fano weighted complete intersections.