  We consider four classes of polynomials over the fields $\mathbb{F}_{q^3}$,
$q=p^h$, $p>3$, $f_1(x)=x^{q^2+q-1}+Ax^{q^2-q+1}+Bx$,
$f_2(x)=x^{q^2+q-1}+Ax^{q^3-q^2+q}+Bx$, $f_3(x)=x^{q^2+q-1}+Ax^{q^2}-Bx$,
$f_4(x)=x^{q^2+q-1}+Ax^{q}-Bx$, where $A,B \in \mathbb{F}_q$. We determine
conditions on the pairs $(A,B)$ and we give lower bounds on the number of pairs
$(A,B)$ for which these polynomials permute $\mathbb{F}_{q^3}$.
