Tauberian conditions for the (C, ?) integrability of functions

For a real-valued continuous function f(x) on [ 0 , ?) , we define (Fomula presented.) for x> 0. We say that ?0?f(u)du is (C, ?) integrable to L for some ?> - 1 if the limit lim x › ???(x) = L exists. It is known that lim x › ?s(x) = L implies lim x › ???(x) = L for all ?> - 1. The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the (C, ?) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the (C, ?) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results. © 2016, Springer International Publishing.