Alternative proofs of some classical Tauberian theorems for the weighted mean method of integrals

Abstract

Let 0 ? p(x) be a nondecreasing real valued differentiable function on [0,?) such that p(0) = 0 and p(x)? ? as x ? ?. Given a real valued function f (x) which is continuous on [0,?) and s(x) = x?0 f(t)dt. We define the weighted mean of s(x) as ?p(x) = 1/p(x) x?0 p'(t)s(t)dt, where p'(t) is derivative of p(t). It is known that if the limit limx?? s(x)=s exists, then limx?? ?p(x) = s also exists. However, the converse is not always true. Adding some suitable conditions to existence of lim x?? ?p(x) which are called Tauberian conditions may imply convergence of the integral ??0 f (t)dt. In this work, we give some classical type Tauberian theorems to retrieve convergence of s(x) out of weighted mean integrability of s(x) with some Tauberian conditions.