Tauberian theorems for the logarithmic summability methods of integrals (Q2414021)

Let $f$ be a real-valued function which is continuous on $[1,\infty]$ and $s(x)=\int_{1}^{x}f(t)\,dt$. The logarithmic mean of $s(x)$ is given by \[ \ell(x)=\frac{1}{\log x}\int_{1}^{x}\frac{s(t)}{t}\,dt. \] If $\lim_{x\to\infty}\ell(x)=s$, then $\int_{1}^{\infty}f(t)\,dt$ is said to be logarithmically summable to~$s$. In this paper, the authors introduce the concept of general logarithmic control modulus for the oscillatory behavior of functions. They prove Tauberian theorems for the logarithmic summability of integrals by imposing restrictions on this new concept.