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We prove that all smooth Fano threefolds in the families 
and 
are K-stable, and we also prove that smooth Fano threefolds in the family 
that satisfy one very explicit generality condition are K-stable.
$\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ OF DEGREE 
$(1,1,2)$
We prove that every smooth divisor in 
$\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree 
$(1,1,2)$ is K-stable.
We show that any 
$n$-dimensional Fano manifold 
$X$ with 
$\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and 
$n\geqslant 2$ is K-stable, where 
$\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of 
$X$ introduced by Tian. In particular, any such 
$X$ admits Kähler–Einstein metrics and the holomorphic automorphism group 
$\operatorname{Aut}(X)$ of 
$X$ is finite.
$\mathbb{Q}$-Fano varieties
The notion of Berman–Gibbs stability was originally introduced by Berman for 
$\mathbb{Q}$-Fano varieties 
$X$. We show that the pair 
$(X,-K_{X})$ is K-stable (respectively K-semistable) provided that 
$X$ is Berman–Gibbs stable (respectively semistable).
