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Klesov, O. I.; Steinebach, J. G.

On preserving the limit points of corresponding objects. (English) Zbl 1442.26003

J. Math. Anal. Appl. 486, No. 2, Article ID 123916, 8 p. (2020).
Summary: Suppose that, for two given sequences \(\{a_n\}\) and \(\{b_n\},\) \[ \liminf\limits_{n \to \infty} \frac{b_n}{a_n} = 1 \] and let a function \(f\) be given. What can then be said about the limit behavior of the corresponding ratio \[ \frac{f(b_n)}{f(a_n)} \] as \(n \to \infty\)? In general, no definite answer can be given to this question. We study a case where a definite answer is possible, namely the case of a regularly varying function \(f\) of nonzero order.

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
40A05 Convergence and divergence of series and sequences

Keywords:

regularly varying functions; slowly varying functions; functions preserving the asymptotic equivalence; corresponding objects; set of limit points
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