Slope inequality for an arbitrary divisor

Let [Formula: see text] be a surjective morphism with connected fibers from a smooth complex projective surface [Formula: see text] to a smooth complex projective curve [Formula: see text] with general fiber [Formula: see text]. In this paper, we develop a more general version of the slope inequality for data [Formula: see text], where [Formula: see text] is an arbitrary relatively effective divisor on [Formula: see text] and [Formula: see text] is a locally free sub-sheaf of [Formula: see text]. We see how the speciality of [Formula: see text], restricted to the general fiber, plays a role in the results. Moreover, we compute some natural examples and provide applications.