  Let $S$ be a smooth minimal surface of general type with a (rational) pencil
of hyperelliptic curves of minimal genus $g$. We prove that if
$K_S^2<4\chi(\mathcal O_S)-6,$ then $g$ is bounded. The surface $S$ is
determined by the branch locus of the covering $S\rightarrow S/i,$ where $i$ is
the hyperelliptic involution of $S.$ For $K_S^2<3\chi(\mathcal O_S)-6,$ we show
how to determine the possibilities for this branch curve. As an application,
given $g>4$ and $K_S^2-3\chi(\mathcal O_S)<-6,$ we compute the maximum value
for $\chi(\mathcal O_S).$ This list of possibilities is sharp.
