K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

We prove that the alpha invariant of a quasi-smooth Fano 3-fold weighted hypersurface of index $1$ is greater than or equal to $1/2$. Combining this with the result of Stibitz and Zhuang \cite{SZ19} on a relation between birational superrigidity and K-stability, we prove the K-stability of a birationally superrigid quasi-smooth Fano 3-fold weighted hypersurfaces of index $1$.