On generalized absolute Cesàro summability factors

Abstract

In this paper, a known theorem dealing with $|C,\alpha ,\gamma ;\delta {|}_k$ summability factors has been generalized for $|C,\alpha ,\beta ,\gamma ;\delta {|}_k$ summability factors. Some results have also been obtained.

MSC:40D15, 40F05, 40G99.

1 Introduction

A sequence $(b_n)$ of positive numbers is said to be quasi-monotone if $n\mathrm{\Delta }{b}_n\ge -\rho b_n$ for some $\rho >0$ and is said to be δ-quasi-monotone, if $b_n\to 0$, $b_n>0$ ultimately and $\mathrm{\Delta }{b}_n\ge -{\delta }_n$, where $({\delta }_n)$ is a sequence of positive numbers (see [1]). Let $\sum a_n$ be a given infinite series with partial sums $(s_n)$. We denote by $u_n^{\alpha ,\beta }$ and $t_n^{\alpha ,\beta }$ the n th Cesàro means of order $(\alpha ,\beta )$, with $\alpha +\beta >-1$, of the sequences $(s_n)$ and $(n{a}_n)$, respectively, that is (see [2]),

(1)

(2)

where

$$A_n^{\alpha +\beta }=O\left(n^{\alpha +\beta }\right),\alpha +\beta >-1,A_0^{\alpha +\beta }=1\text{and}{A}_{-n}^{\alpha +\beta }=0\text{for }n>0.$$

(3)

The series $\sum a_n$ is said to be summable $|C,\alpha ,\beta {|}_k$, $k\ge 1$ and $\alpha +\beta >-1$, if (see [3])

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{k-1}|u_n^{\alpha ,\beta }-u_{n-1}^{\alpha ,\beta }{|}^k<\mathrm{\infty }.$$

(4)

Since $t_n^{\alpha ,\beta }=n(u_n^{\alpha ,\beta }-u_{n-1}^{\alpha ,\beta })$ (see [3]), condition (4) can also be written as

$$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}|t_n^{\alpha ,\beta }{|}^k<\mathrm{\infty }.$$

(5)

The series $\sum a_n$ is said to be summable $|C,\alpha ,\beta ,\gamma ;\delta {|}_k$, $k\ge 1$, $\alpha +\beta >-1$, $\delta \ge 0$ and γ is a real number, if (see [4])

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{\gamma (\delta k+k-1)}|u_n^{\alpha ,\beta }-u_{n-1}^{\alpha ,\beta }{|}^k=\sum _{n=1}^{\mathrm{\infty }}{n}^{\gamma (\delta k+k-1)-k}|t_n^{\alpha ,\beta }{|}^k<\mathrm{\infty }.$$

(6)

If we take $\beta =0$, then $|C,\alpha ,\beta ,\gamma ;\delta {|}_k$ summability reduces to $|C,\alpha ,\gamma ;\delta {|}_k$ summability (see [5]).

2 Known result

In [6], we have proved the following theorem dealing with $|C,\alpha ,\gamma ;\delta {|}_k$ summability factors of infinite series.

Theorem A Let $k\ge 1$, $0\le \delta <\alpha \le 1$, and γ be a real number such that $-\gamma (\delta k+k-1)+(\alpha +1)k>1$. Suppose that there exists a sequence of numbers $(B_n)$ such that it is δ-quasi-monotone with $|\mathrm{\Delta }{\lambda }_n|\le |B_n|$, ${\lambda }_n\to 0$ as $n\to \mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}n{\delta }_{n}logn<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}n{B}_{n}logn$ is convergent. If the sequence $(w_n^{\alpha })$ defined by (see [7])

(7)

(8)

satisfies the condition

$$\sum _{n=1}^m{n}^{\gamma (\delta k+k-1)-k}{\left(w_n^{\alpha }\right)}^k=O(logm)\mathit{\text{as}}m\to \mathrm{\infty },$$

(9)

then the series $\sum a_n{\lambda }_n$ is summable $|C,\alpha ,\gamma ;\delta {|}_k$.

3 The main result

The aim of this paper is to generalize Theorem A for $|C,\alpha ,\beta ,\gamma ;\delta {|}_k$ summability. We shall prove the following theorem.

Theorem Let $k\ge 1$, $0\le \delta <\alpha \le 1$, and γ be a real number such that $(\alpha +\beta +1-\gamma (\delta +1))k>1$, and let there be sequences $(B_n)$ and $({\lambda }_n)$ such that the conditions of Theorem A are satisfied. If the sequence $(w_n^{\alpha ,\beta })$ defined by

(10)

(11)

satisfies the condition

$$\sum _{n=1}^m{n}^{\gamma (\delta k+k-1)-k}{\left(w_n^{\alpha ,\beta }\right)}^k=O(logm)\mathit{\text{as}}m\to \mathrm{\infty },$$

(12)

then the series $\sum a_n{\lambda }_n$ is summable $|C,\alpha ,\beta ,\gamma ;\delta {|}_k$. It should be noted that if we take $\beta =0$, then we get Theorem A.

We need the following lemmas for the proof of our theorem.

Lemma 1 ([8])

Under the conditions on $(B_n)$, as taken in the statement of the theorem, we have the following:

(13)

(14)

Lemma 2 ([9])

If $0<\alpha \le 1$, $\beta >-1$, and $1\le v\le n$, then

$$\left|\sum _{p=0}^v{A}_{n-p}^{\alpha -1}{A}_p^{\beta }{a}_p\right|\le \underset{1\le m\le v}{max}\left|\sum _{p=0}^m{A}_{m-p}^{\alpha -1}{A}_p^{\beta }{a}_p\right|.$$

(15)

4 Proof of the theorem

Let $(T_n^{\alpha ,\beta })$ be the n th $(C,\alpha ,\beta )$ mean of the sequence $(n{a}_n{\lambda }_n)$. Then by (2), we have

$$T_n^{\alpha ,\beta }=\frac{1}{A_n^{\alpha +\beta }}\sum _{v=1}^n{A}_{n-v}^{\alpha -1}{A}_v^{\beta }v{a}_v{\lambda }_v.$$

Firstly applying Abel’s transformation and then using Lemma 2, we have that

since

$$|T_{n,1}^{\alpha ,\beta }+T_{n,2}^{\alpha ,\beta }{|}^k\le 2^k\left(\right|T_{n,1}^{\alpha ,\beta }{|}^k+\left|T_{n,2}^{\alpha ,\beta }{|}^k\right).$$

(16)

In order to complete the proof of the theorem, by (6), it is sufficient to show that for $r=1,2$,

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{\gamma (\delta k+k-1)-k}|T_{n,r}^{\alpha ,\beta }{|}^k<\mathrm{\infty }.$$

Whenever $k>1$, we can apply Hölder’s inequality with indices k and $k^{\prime }$, where $\frac{1}{k}+\frac{1}{k^{\prime }}=1$, we get that

in view of the hypotheses of the theorem and Lemma 1. Similarly, we have that

$$\begin{array}{rcl}\sum _{n=2}^{m+1}{n}^{\gamma (\delta k+k-1)-k}|T_{n,2}^{\alpha ,\beta }{|}^k& =& O(1)\sum _{n=1}^m|{\lambda }_n|n^{\gamma (\delta k+k-1)-k}{\left(w_n^{\alpha ,\beta }\right)}^k\\ =& O(1)\sum _{n=1}^{m-1}|\mathrm{\Delta }{\lambda }_n|\sum _{v=1}^n{v}^{\gamma (\delta k+k-1)-k}{\left(w_v^{\alpha ,\beta }\right)}^k\\ +O(1)|{\lambda }_m|\sum _{v=1}^m{v}^{\gamma (\delta k+k-1)-k}{\left(w_v^{\alpha ,\beta }\right)}^k\\ =& O(1)\sum _{n=1}^{m-1}|\mathrm{\Delta }{\lambda }_n|logn+O(1)|{\lambda }_m|logm\\ =& O(1)\sum _{n=1}^{m-1}|B_n|logn+O(1)|{\lambda }_m|logm\\ =& O(1)\text{as }m\to \mathrm{\infty },\end{array}$$

by virtue of the hypotheses of the theorem and Lemma 1. Therefore, by (6), we get that for $r=1,2$,

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{\gamma (\delta k+k-1)-k}|T_{n,r}^{\alpha ,\beta }{|}^k<\mathrm{\infty }.$$

This completes the proof of the theorem. If we take $\delta =0$ and $\gamma =1$, then we get a result for $|C,\alpha ,\beta {|}_k$ summability factors. Also, if we take $\beta =0$, $\delta =0$, and $\alpha =1$, then we get a result for $|C,1{|}_k$ summability.