Some classical Tauberian theorems for (C,1,1,1) summable triple sequences

Let (umns) be a (C,1,1,1) summable triple sequence of real numbers. We give one-sided Tauberian conditions of Landau and Hardy type under which (umns) converges in Pringsheim's sense. We prove that (umns) converges in Pringsheim's sense if (umns) is slowly oscillating in certain senses. Moreover, we extend a Tauberian theorem given by Móricz [11] for double sequences to triple sequences