  Let $X$ be a surface of general type with maximal Albanese dimension: if
$K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq
4\chi(\mathcal{O}_X)+4(q-2)$. We give a complete classification of surfaces for
which equality holds for $q(X)\geq 3$: these are surfaces whose canonical model
is a double cover of a product elliptic surface branched over an ample divisor
with at most negligible singularities which intersects the elliptic fibre
twice. We also prove, in the same hypothesis, that a surface $X$ with
$K_X^2\neq 4\chi(\mathcal{O}_X)+4(q-2)$ satisfies $K_X^2\geq
4\chi(\mathcal{O}_X)+8(q-2)$ and we give a characterization of surfaces for
which the equality holds. These are surfaces whose canonical model is a double
cover of an isotrivial smooth elliptic surface branched over an ample divisor
with at most negligible singularities whose intersection with the elliptic
fibre is $4$.
