  Given a relatively minimal fibration $f: S \to \Bbb P^1$ on a rational
surface $S$ with general fiber $C$ of genus $g$, we investigate under what
conditions the inequality $6(g-1)\le K_f^2$ occurs, where $K_f$ is the
canonical relative sheaf of $f$. We give sufficient conditions for having such
inequality, depending on the genus and gonality of $C$ and the number of
certain exceptional curves on $S$. We illustrate how these results can be used
for constructing fibrations with the desired property. For fibrations of genus
$11\le g\le 49$ we prove the inequality: $$ 6(g-1) +4 -4\sqrt g \le K_f^2.$$
