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We define a new class of positive and Lebesgue measurable functions in terms of their asymptotic behavior, which includes the class of regularly varying functions. We also characterize it by transformations, corresponding to generalized moments when these functions are random variables. We study the properties of this new class and discuss their applications to Extreme Value Theory.

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Very recently Seneta [15] has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each y > 0,We extend this result by considering a wider class of functions and a more general condition than (1). Further, a representation theorem for this wider class is provided.