Tauberian theorems concerning weighted mean summable integrals

Abstract

Let p be a positive real-valued continuous function on R+ such that the function P(x) = integral(x)(0) p(t) dt, x > 0, is regularly varying with a positive index in the Karamata sense. For a real- or complex-valued continuous function f on R+, we define s(x) = integral(x)(0) f (y) dy and sigma(p)( x) = 1/P(x) integral(x)(0) s(y) p(y) dy. It is known that if the finite limit lim(x ->infinity) s(x) = L exists, then so does lim(x ->infinity) sigma(p)(x) = L. In this paper, we introduce some Tauberian conditions in terms of the weighted classical control modulo and the weighted general control modulo of order one under which the converse implication and its extensions hold. Our results generalize some classical type Tauberian theorems existing in the literature.