﻿Abstract
The $(C,\alpha)$ Integrability of Functions by Weighted Mean Methods

İbrahim Çanak, Ümit Totur

Let $p(x)$ be a nondecreasing continuous function on $[0,\infty)$ such that $p(0)=0$ and $p(t)\to \infty$ as $t\to \infty$. For a continuous function $f(x)$ on $[0,\infty)$, we define\\ $s(t)=\int_{0}^{t} f(u)du$ and $\sigma_{\alpha}(t)=\int_{0}^{t}(1-\frac{p(u)}{p(t)})^{\alpha}f(u)du.$\\ We say that a continuous function $f(x)$ on $[0,\infty)$ is $(C,\alpha)$ integrable to a by the weighted mean method determined by the function $p(x)$ for some $\alpha> -1$ if the limit $lim_{t\to \infty} \sigma_{\alpha}(t)=a$ exists. We prove that if the limit $lim_{t\to \infty} \sigma_{\alpha}(t)=a$ exists for some $\alpha>-1$, then the limit $lim_{t\to \infty} \sigma_{\alpha +h}(t)=a$ exists for all $h>0$. Next, we prove that if the limit $lim_{t\to \infty} \sigma_{\alpha}(t)=a$ exists for some $\alpha >0$ and\\ $\frac{p(t)}{p'(t)}f(t)=O(1)$, $t\to \infty$,\\ then the limit $lim_{t\to \infty} \sigma_{\alpha -1}(t)=a$ exist. Marcel G. De Bruin 664