Some Tauberian Conditions for the Logarithmic Summability Method of Integrals

Abstract

Let $f$ be a real valued continuous function on $[1, \infty)$ and $s(x)=\int_{1}^{x}f(t)dt$. The logarithmic mean of $s(x)$ is defined by $\ell(x)=\frac{1}{\log{x}}\int_1^x\frac{s(t)}{t}dt.$ If the limit $\displaystyle{\lim_{x \to \infty}\ell(x)=\alpha}$ exists, then we say that the improper integral $\int_1^{\infty}f(t)dt$ is summable by logarithmic summability method to a finite number $\alpha$. It is known that if the improper integral $\int_1^{\infty}f(t)dt$ is summable, then it is also summable by logarithmic summability method to same limit. However the converse implication is not always true. In this paper, we give the concept of slow oscillation with respect to logarithmic summability method and general logarithmic control modulo. Our goal is to obtain some Tauberian theorems for the logarithmic summability method of integrals by using these concepts.