  Let $X$ be a surface of general type with maximal Albanese dimension over an
algebraically closed field of characteristic greater than two: we prove that if
$K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq
4\chi(\mathcal{O}_X)+4(q-2)$. Moreover 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. In addition we expose a similar partial result over
algebraically closed fields of characteristic two.
  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 $4$.
