Journal of Inequalities and Applications
Volume 1 (1997), Issue 3, Pages 253-274

𝒪-regularly varying functions in approximation theory

Stefan Jansche

Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, Kiel D-24098, Germany

Received 10 June 1996

Copyright © 1997 Stefan Jansche. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For 𝒪-regularly varying functions a growth relation is introduced and characterized which gives an easy tool in the comparison of the rate of growth of two such functions at the limit point. In particular, methods based on this relation provide necessary and sufficient conditions in establishing chains of inequalities between functions and their geometric, harmonic, and integral means, in both directions. For periodic functions, for example, it is shown how this growth relation can be used in approximation theory in order to establish equivalence theorems between the best approximation and moduli of smoothness from prescribed inequalities of Jackson and Bernstein type.