In this paper, we prove Tauberian theorems of slowly oscillating type for the (J, p) summability method by using a result due to Mikhalin [G.A. Mikhalin, Theorem of Tauberian type for (J, pn) summation methods, Ukrain. Mat. Zh. 29 (1977), 763-770. English translation: Ukrain. Math. J. 29 (1977) 564-5691. (c) 2011 Elsevier Ltd. All rights reserved.