Abstract: Let T be a closed positive current of dimension p in and H an analytic subvariety of of dimension . We denote by the slice of T in H if it exists. We consider the functions on defined by and . The aim of this paper is to establish relationship between the growth of these functions. We prove results generalizing those of L. Gruman in [Séminaire Lelong–Skoda 1981–1983, Lecture Notes in Math., vol. 1028, Springer-Verlag, Berlin, 1983, pp. 125–162] and B. Molzan, B. Shiffman and N. Sibony in [Math. Ann. 257 (1981) 43]. [Copyright &y& Elsevier]