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

Abstract

Let $0 \not\neq p(x) $ be a nondecreasing real valued differentiable function on $[0, \infty)$ such that $p(0)=0$ and $p(x) \rightarrow \infty $ as $x \to \infty $.
Given a real valued function $f(x)$ which is continuous on $[0, \infty)$ and
$$
s(x)=\int_0^x f(t) dt.
$$
We define the weighted mean of $s(x)$ as
$$
\sigma_{p}(x)=\frac{1}{p(x)}\int_0^x p'(t)s(t) dt,
$$
where $p'(t)$ is derivative of $p(t).$ It is known that if the limit $\displaystyle{\lim_{x \rightarrow \infty}s(x)=s}$ exists, then $\displaystyle{\lim_{x \to \infty} \sigma_p (x)=s}$ also exists. However, the converse is not always true.
Adding some suitable conditions to existence of $\displaystyle{\lim_{x \rightarrow \infty} \sigma_p (x)}$ which are called Tauberian
conditions may imply convergence of the integral $\int_0^{\infty} f(t)dt$.

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.