**4. Concluding Remarks**

In this paper, new results on the condition, for some functions *r* and *g*,

$$\lim\_{x \to \infty} r(x) \left( \frac{\mathcal{U}(x + y\mathcal{g}(x))}{\mathcal{U}(x)} - c^{\mathcal{A}y} \right) = \theta(y), \quad \forall y.$$

where we assume that the convergence is l.u. in *y*, are presented. This limit generalizes the ones analyzed by Seneta [4] and Omey and Cadena [5], both of them being related to the monotony of functions in the Zygmund sense. Under this analysis, properties of *<sup>θ</sup>*(*y*) are described. Representations of the functions *U* involved in this limit are provided.

**Author Contributions:** The authors have equally contributed to the writing, editing and style of the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
