On the slope of relatively minimal fibrations on rational complex surfaces

Let $S$ be a complex rational surface and consider a relatively minimal fibration $f:S\to\Bbb P^1$ with general fiber a curve $C$ of genus $g$. We investigate under what conditions the inequality
$6(g-1)\leq K^2_f$ occurs, where $K^2_f$ is the canonical relative sheaf of
$f$. We give conditions for having such inequality, depending of the genus and
gonality of $C$ and the number of certain exceptional curves on $S$. We construct examples of fibrations with the desired properties.