A sharp bound for the slope of double cover fibrations

Let $f\colon X\rightarrow B$ be a fibration of genus $g$ whose general fiber is a double cover of a smooth curve of genus $\gamma$. We show that $4(g-1)/(g-\gamma)$ is a sharp lower bound for the slope of $f$ when $g> 4\gamma+1$, proving a conjecture of Barja. Moreover, we give a characterization of the fibered surfaces that reach the bound. In the case $g=4\gamma+1 $ we obtain the same sharp bound under the additional assumption that the involutions on the general fibers glue to a global involution on $X$.