  We prove that a complex surface S with irregularity q(S)=5 that has no
irrational pencil of genus >1 has geometric genus p_g(S)>7. As a consequence,
one is able to classify minimal surfaces S of general type with q(S)=5 and
p_g(S)<8. This result is a negative answer, for q=5, to the question asked in
arXiv:0811.0390 of the existence of surfaces of general type with irregularity
q>3 that have no irrational pencil of genus >1 and with the lowest possible
geometric genus p_g=2q-3. This gives some evidence for the conjecture that the
only irregular surface with no irrational pencil of genus >1 and p_g=2q-3 is
the symmetric product of a genus three curve.
