O-regularly varying functions in approximation theory

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.