One-sided Tauberian conditions for a general summability method

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