Strict Arakelov inequality for a family of varieties of general type

Let f: X Y be a semistable non-isotrivial family of n-folds over a smooth projective curve with discriminant locus S (Formula presented) Y and with general fiber F of general type. We show the strict Arakelov inequality (Formula presented) for all v (Formula presented) (Formula presented) such that the v-th pluricanonical linear system |(Formula presented)| is birational. This answers a question asked by Möller, Viehweg and the third named author [1].