  In this paper we investigate the numerical properties of relatively minimal
isotrivial fibrations $\varphi \colon X \lr C$, where $X$ is a smooth,
projective surface and $C$ is a curve. In particular we prove that, if $g(C)
\geq 1$ and $X$ is neither ruled nor isomorphic to a quasi-bundle, then $K_X^2
\leq 8 \chi(\mO_X)-2$; this inequality is sharp and if equality holds then $X$
is a minimal surface of general type whose canonical model has precisely two
ordinary double points as singularities. Under the further assumption that
$K_X$ is ample, we obtain $K_X^2 \leq 8 \chi(\mO_X)-5$ and the inequality is
also sharp. This improves previous results of Serrano and Tan.
