A theorem of Galambos-Bojanić-Seneta type. (English) Zbl 1168.26300

Summary: In the theorems of Galambos-Bojanić-Seneta’s type, the asymptotic behavior of the functions \(c_{[x]},\, x\geq 1\), for \(x\rightarrow +\infty\), is investigated by the asymptotic behavior of the given sequence of positive numbers \((c_{n})\), as \(n\rightarrow +\infty \) and vice versa. The main result of this paper is one theorem of such a type for sequences of positive numbers \((c_{n})\) which satisfy an asymptotic condition of the Karamata type \(\underline{\lim}_{n \to \infty} c_{[\lambda n]}/c_n > 1\), for \(\lambda >1\).


26A12 Rate of growth of functions, orders of infinity, slowly varying functions
26A48 Monotonic functions, generalizations
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[1] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, vol. 27 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1987. · Zbl 0617.26001
[2] D. B. H. Cline, “Intermediate regular and \Pi variation,” Proceedings of the London Mathematical Society, vol. 68, no. 3, pp. 594-616, 1994. · Zbl 0793.26004
[3] D. Djur, “\?-regularly varying functions and strong asymptotic equivalence,” Journal of Mathematical Analysis and Applications, vol. 220, no. 2, pp. 451-461, 1998. · Zbl 0920.26004
[4] D. Arandjelović, “\?-regular variation and uniform convergence,” Publications de l/Institut Mathématique, vol. 48(62), pp. 25-40, 1990. · Zbl 0731.26004
[5] S. M. Berman, “Sojourns and extremes of a diffusion process on a fixed interval,” Advances in Applied Probability, vol. 14, no. 4, pp. 811-832, 1982. · Zbl 0494.60076
[6] D. Djur and A. Torga\vsev, “Strong asymptotic equivalence and inversion of functions in the class Kc,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 383-390, 2001. · Zbl 0991.26002
[7] B. I. Korenblum, “On the asymptotic behavior of Laplace integrals near the boundary of a region of convergence,” Doklady Akademii Nauk SSSR, vol. 104, pp. 173-176, 1955 (Russian).
[8] W. Matuszewska, “On a generalization of regularly increasing functions,” Studia Mathematica, vol. 24, pp. 271-279, 1964. · Zbl 0171.33503
[9] U. Stadtmüller and R. Trautner, “Tauberian theorems for Laplace transforms,” Journal für die Reine und Angewandte Mathematik, vol. 311-312, pp. 283-290, 1979. · Zbl 0409.44003
[10] E. Seneta, Functions of Regular Variation, vol. 506 of LNM, Springer, New York, NY, USA, 1976. · Zbl 0324.26002
[11] R. Schmidt, “Über divergente Folgen und lineare Mittelbildungen,” Mathematische Zeitschrift, vol. 22, no. 1, pp. 89-152, 1925. · JFM 51.0182.04
[12] D. E. Grow and \vC. V. Stanojević, “Convergence and the Fourier character of trigonometric transforms with slowly varying convergence moduli,” Mathematische Annalen, vol. 302, no. 3, pp. 433-472, 1995. · Zbl 0827.42003
[13] D. Djur and A. Torga\vsev, “Representation theorems for sequences of the classes CRc and ERc,” Siberian Mathematical Journal, vol. 45, no. 5, pp. 834-838, 2004.
[14] \vC. Stanojević, “Structure of Fourier and Fourier-Stieltjes coefficients of series with slowly varying convergence moduli,” Bulletin of the American Mathematical Society, vol. 19, no. 1, pp. 283-286, 1988. · Zbl 0663.42008
[15] D. Djur and A. Torga\vsev, “\?-regular variability and power series,” Filomat, no. 15, pp. 215-220, 2001. · Zbl 1059.26003
[16] R. Bojanic and E. Seneta, “A unified theory of regularly varying sequences,” Mathematische Zeitschrift, vol. 134, pp. 91-106, 1973. · Zbl 0256.40002
[17] J. Karamata, Theory and Practice of the Stieltjes Integral, vol. 154, Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia, 1949.
[18] J. Galambos and E. Seneta, “Regularly varying sequences,” Proceedings of the American Mathematical Society, vol. 41, no. 1, pp. 110-116, 1973. · Zbl 0247.26002
[19] L. de Haan, On Regular Variation and Its Application to the Weak Convergence of Sample Extremes, vol. 32 of Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam, The Netherlands, 1970. · Zbl 0226.60039
[20] W. Matuszewska and W. Orlicz, “On some classes of functions with regard to their orders of growth,” Studia Mathematica, vol. 26, pp. 11-24, 1965. · Zbl 0134.31604
[21] D. Djur, Lj. D. R. Ko\vcinac, and M. R. \vZi, “On selection principles and games in divergente processes,” in Selection Principles and Covering Properties in Topology, Lj. D. R. Ko\vcinac, Ed., Quaderni di Matematica 18, Seconda Università di Napoli, Caserta, Italy, 2006.
[22] D. Djur, Lj. D. R. Ko\vcinac, and M. R. \vZi, “Rapidly varying sequences and rapid convergence,” Topology and Its Applications, vol. 155, no. 17-18, pp. 2143-2149, 2008. · Zbl 1154.54002
[23] D. Drasin and E. Seneta, “A generalization of slowly varying functions,” Proceedings of the American Mathematical Society, vol. 96, no. 3, pp. 470-472, 1986. · Zbl 0591.26004
[24] D. Djur and V. Bo\vzin, “A proof of S. Aljan hypothesis on \?-regularly varying sequences,” Publications de l/Institut Mathématique, vol. 61, no. 76, pp. 46-52, 1997.
[25] D. Djur, “A theorem on a representation of \ast -regularly varying sequences,” Filomat, no. 16, pp. 1-6, 2002. · Zbl 1082.26002
[26] D. Djur, Lj. D. R. Ko\vcinac, and M. R. \vZi, “Some properties of rapidly varying sequences,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 1297-1306, 2007. · Zbl 1116.26002
[27] D. Djur and A. Torga\vsev, “On the Seneta sequences,” Acta Mathematica Sinica, vol. 22, no. 3, pp. 689-692, 2006. · Zbl 1170.26300
[28] D. Djur, A. Torga\vsev, and S. Je, “The strong asymptotic equivalence and the generalized inverse,” Siberian Mathematical Journal, vol. 49, no. 4, pp. 628-636, 2008.
[29] V. V. Buldygin, O. I. Klesov, and J. G. Steinebach, “On some properties of asymptotically quasi-inverse functions and their application-I,” Theory of Probability and Mathematical Statistics, no. 70, pp. 11-28, 2004. · Zbl 1103.26001
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