On the slope conjecture of Barja and Stoppino for fibred surfaces

Let $f:,S to B$ be a locally non-trivial relatively minimal fibration of genus $ggeq 2$ with relative irregularity $q_f$. It was conjectured by Barja and Stoppino that the slope $lambda_fgeq frac{4(g-1)}{g-q_f}$. We prove the conjecture when $q_f$ is small with respect to $g$; we also construct counterexamples when $g$ is odd and $q_f=(g+1)/2$.