  Let $X$ be a minimal surface of general type and maximal Albanese dimension
with irregularity $q\geq 2$. We show that $K_X^2\geq 4\chi(\mathcal
O_X)+4(q-2)$ if $K_X^2<\frac92\chi(\mathcal O_X)$, and also obtain the
characterization of the equality. As a consequence, we prove a conjecture of
Manetti on the geography of irregular surfaces if $K_X^2\geq 36(q-2)$ or
$\chi(\mathcal O_X)\geq 8(q-2)$, and we also prove a conjecture that surfaces
of general type and maximal Albanese dimension with $K_X^2=4\chi(\mathcal O_X)$
are exactly the resolution of double covers of abelian surfaces branched over
ample divisors with at worst simple singularities.
