General Tauberian theorems for the Cesàro integrability of functions

###EgeUn###For a locally integrable function f on [ 0, ?), we define F(t) = ? t 0 f(u)du and ? ? (t) = ? t 0 (1 - u t) ? (u)du for t ;gt; 0. The improper integral ? ? 0 f(u)du is said to be (C, ?) integrable to L for some ? ;gt; - 1 if the limit lim x › ? ? ?(t) = L exists. It is known that lim t › ? F(t) = l implies lim t › ? ? ?(t) = l for ? ;gt; - 1, but the converse of this implication is not true in general. In this paper, we introduce the concept of the general control modulo of non-integer order for functions and obtain some Tauberian conditions in terms of this concept for the (C, ?) integrability method in order that the converse implication hold true. Our results extend the main theorems in [Ü. Totur and I. Çanak, Tauberian conditions for the (C, ?) integrability of functions, Positivity 21 2017, 1, 73-83]. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019