One-sided Tauberian conditions for (C,1)(C,1) summability method of integrals

Let ff be a real valued function which is continuous on [0,∞)[0,∞). In this paper we prove a Tauberian-like theorem to retrieve slow oscillation of s(x)=∫0xf(t)dt out of its (C,1)(C,1) summability of the generator function v1(x)v1(x) and some additional conditions. As a corollary we recover convergence of s(x)s(x).