A k-hypertournament is a complete k-hypergraph with each k-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a k-hypertournament, the score siβ (losing score riβ) of a vertex viβ is the number of arcs containing viβ in which viβ is not the last element (in which viβ is the last element). The total score of viβ is defined as tiβ=siββriβ. In this paper we obtain stronger inequalities for the quantities βiβIβriβ, βiβIβsiβ and βiβIβtiβ, where Iβ{1,2,β¦,n}. Furthermore, we discuss the case of equality for these inequalities. We also characterize total score sequences of strong k-hypertournaments