Statistical extension of classical Tauberian theorems in the case of logarithmic summability of locally integrable functions on $[1,\mathrm{\infty })$

Let $s:[1,\mathrm{\infty })\to \text{C}$ be a locally integrable function in Lebesgue’s sense. The logarithmic (also called harmonic) mean of the function $s$ is defined by [\tau(t) := \frac 1{\log t} \int_1^t \frac {s(x)}{x} dx, \qquad t>1,] where the logarithm is to base $e$. Besides the ordinary limit ${\lim }_{x\to \mathrm{\infty }}s(x)$, we also use the notion of the so-called statistical limit of $s$ at $\mathrm{\infty }$, in notation: ${\text{stlim}}_{x\to \mathrm{\infty }}s(x)=\mathrm{\ell }$, by which we mean that for every $\text{e}>0$, [\lim_{b\to \infty} \frac 1b \Big | \Big {x\in(1,b): |s(x)-\ell| >\e \Big} \Big| = 0.] We also use the ordinary limit ${\lim }_{t\to \mathrm{\infty }}\tau (t)$ as well as the statistical limit ${\text{stlim}}_{t\to \mathrm{\infty }}\tau (t)$. We will prove the following Tauberian theorem: Suppose that the real-valued function $s$ is slowly decreasing or the complex-valued $s$ is slowly oscillating. If the statistical limit ${\text{stlim}}_{t\to \mathrm{\infty }}\tau (t)=\mathrm{\ell }$ exists, then the ordinary limit ${\lim }_{x\to \mathrm{\infty }}s(x)=\mathrm{\ell }$ also exists.