One-sided Tauberian conditions for a general summability method

Let (un) be a sequence of real numbers and L be an additive summability method with some property. We show that if slow decrease of (un) or one-sided boundedness of the classical control modulo of the oscillatory behavior of (un) is a Tauberian condition for a general summability method L, then one-sided boundedness by a sequence with certain conditions of the general control modulo of the oscillatory behavior of integer order m of (un) is also a Tauberian condition for L. © 2011 Elsevier Ltd. All rights reserved.