For a real-valued continuous function
f(
x) on
([0,infty )), we define
$$begin{aligned} s(x)=int _{0}^{x} f(u)duquad text {and}quad sigma _{alpha } (x)= int _{0}^{x}left( 1-frac{u}{x}right) ^{alpha }f(u)du end{aligned}$$
for
(x>0). We say that
(int _{0}^{infty } f(u)du) is
((C, alpha )) integrable to
L for some
(alpha >-1) if the limit
(lim _{x rightarrow infty } sigma _{alpha } (x)=L) exists. It is known that
(lim _{x rightarrow infty } s(x) =L) implies
(lim _{x rightarrow infty }sigma _{alpha } (x) =L) for all
(alpha >-1). The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the
((C, alpha )) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the
((C,alpha )) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results.