TAUBERIAN CONDITIONS FOR $q$-CES\`{A}RO INTEGRABILITY

Given a $q$-integrable function $f$ on $[0, \infty)$, we define $s(x)=\int_{0}^{x}f(t)d_qt$ and $\sigma(s(x))=\frac{1}{x}\int _{0}^{x} s(t)d_{q}t$ for $x>0$. It is known that if $\lim _{x \to \infty}s(x)$ exists andis equal to $A$, then $\lim _{x \to \infty}\sigma(s(x))=A$. But the converse of this implication is not true in general. Our goal is to obtain Tauberian conditions imposed on the general control modulo of $s(x)$ under which the converse implication holds. These conditions generalize some previously obtained Tauberian conditions