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

WOS: 000395063500007For a real-valued continuous function f(x) on , we define s(x) = integral(x)(0) f(u)du and sigma(alpha) (x) = integral(x)(0) (1 - u/x)(alpha) f(u)du for x > 0. We say that integral(infinity)(0) f(u)du is (C, alpha) integrable to L for some alpha > -1 if the limit(x ->infinity) exists. It is known that implies for all . The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results