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Zbl 0554.40008
Bor, Hüseyin
On two summability methods.
(English)
[J] Math. Proc. Camb. Philos. Soc. 97, 147-149 (1985). ISSN 0305-0041; ISSN 1469-8064/e

Let $\sum a\sb n$ be given infinite series with partial sums $s\sb n$, and $\sqrt{n}=na\sb n$. By $U\sb n\sp{\alpha}$ and $t\sb n\sp{\alpha}$ we denote the nth Cesàro means of order $\alpha$ $(\alpha >-1)$ of the sequences $(s\sb n)$ and $(r\sb n)$, respectively. The series $\sum a\sb n$ is said to be $\vert C,\alpha \vert\sb k$, $k\ge 1$, if $\sum\sp{\infty}\sb{n=1}n\sp{k-1}\vert U\sb n\sp{\alpha}- U\sp{\alpha}\sb{n-1}\vert\sp k<\infty$ [see {\it T. M. Flett}, Proc. Lond. Math. Soc. 7, 113-141 (1957; Zbl 0109.044)]. Since $t\sb n\sp{\alpha} = n(U\sb n\sp{\alpha} - U\sp{\alpha}\sb{n-1})$, we can also write $\sum\sp{\infty}\sb{n=1}(1/n)\vert t\sb n\sp{\alpha}\vert\sp k<\infty$. Let $(p\sb n)$ be a sequence of positive real constants such that $P\sb n=p\sb 0+p\sb 1+p\sb 2+...+p\sb n\to \infty$, as $n\to \infty$, $(P\sb{-1}=p\sb{-1}=0)$. Let $T\sb n$ denote the $(\bar N,p\sb n)$ mean of the series $\sum a\sb n$. The series $\sum a\sb n$ is said to be $\vert \bar N,p\sb n\vert\sb k$, $k\ge 1$, if $$\sum\sp{\infty}\sb{n=1}(P\sb n/p\sb n)\sp{k-1}\vert T\sb n - T\sb{n- 1}\vert\sp k<\infty,$$ the author [see J. Univ. Kuwait, Sci. 10, 37-42 (1983; Zbl 0519.40011)]. We prove the following theorem. Theorem. Let $(p\sb n)$ be a sequence of positive real constants such that, as $n\to \infty$ (i) $np\sb n=O(P\sb n)$, (ii) $p\sb n=O(np\sb n)$. If $\sum a\sb n$ is summable $\vert C\vert \vert\sb k$, then it is also summable $\vert \bar N,p\sb{n\vert k,}k\ge 1$.
MSC 2000:
*40F05 Special cases of summability

Keywords: Cesàro means

Citations: Zbl 0109.044; Zbl 0519.40011

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