On the slope of relatively minimal fibrations on rational complex surfaces

Given a relatively minimal fibration $f:S\to {\mathbb{P}}^1$ 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: $$6(g-1)+4-4\sqrt{g}\le K_f^2\mathrm{.}$$