In 2019 Seneta has provided a characterization of slowly varying functions L
in the Zygmund sense by using the condition, for each
. Very recently, we have extended this result by considering a wider class of functions U
related to the following more general condition. For each
, for some functions r
. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.
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