  Let f: X->B be a fibred surface of genus g whose general fibre is a double
cover of a smooth curve of genus gamma. We show that, for g > 4gamma+1, the
number 4(g-1)/(g-gamma) is a sharp lower bound for the slope of f, proving a
conjecture of Barja. Moreover, we give a characterisation of the fibred
surfaces that reach the bound. In the case g = 4gamma+1 we obtain the same
sharp bound under the assumption that the involutions on the general fibres
glue to a global involution on X.
