  We study minimal complex surfaces S of general type with q(S)=q and
p_g(S)=2q-3, q>= 5. We give a complete classification in case that S has a
fibration onto a curve of genus >=2. For these surfaces K^2=8\chi. In general
we prove that K^2>=7\chi-1 and that the stronger inequality K^2\ge 8\chi holds
under extra assumptions (e.g., if the canonical system has no fixed part or the
canonical map has even degree).
  We also describe the Albanese map of S.
