Abstract
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.