*Article* **On Convergence Rates of Some Limits**

#### **Edward Omey 1,\* and Meitner Cadena 2**


Received: 9 March 2020; Accepted: 17 April 2020; Published: 21 April 2020

**Abstract:** In 2019 Seneta has provided a characterization of slowly varying functions *L* in the Zygmund sense by using the condition, for each *y* > 0, *x <sup>L</sup>*(*x*+*y*) *<sup>L</sup>*(*x*) − 1 → 0 as *x* → ∞. Very recently, we have extended this result by considering a wider class of functions *U* related to the following more general condition. For each *y* > 0, *r*(*x*) *<sup>U</sup>*(*x*+*yg*(*x*)) *<sup>U</sup>*(*x*) − 1 → 0 as *x* → <sup>∞</sup>, for some functions *r* and *g*. 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.

**Keywords:** slowly varying; monotony in the Zygmund sense; class <sup>Γ</sup>*a*(*g*); self-neglecting function; convergence rates
