The Index Function Operator for O-regularly Varying Functions

Abstract

The paper examines the functional transformation K of the class ORVφ (see [3]) into the class of positive functions on interval (0, +∞) defined as follows: (0.1) K(f ) = kf , where kf (λ) = lim sup x→+∞ f (λx) f (x) , λ ∈ (0, +∞), and f ∈ ORVφ. Let f ∈ IRVφ or SOφ (see [4]), K be the transformation (0.1) and for any n ∈ N, Kn(f ) = K(K · · · (K ︸ ︷︷ ︸ n (f )) · · · ), then the function p(s) = limn→+∞ Kn(f )(s), s > 0, is IRVφ (and continuous) and SOφ, respectively.