A new application of quasi power increasing sequences. II

Abstract

In this paper, we prove a general theorem dealing with absolute Cesàro summability factors of infinite series by using a quasi-f-power increasing sequence instead of a quasi-σ-power increasing sequence. This theorem also includes several new results.

MSC:26D15, 40D15, 40F05, 40G99, 46A45.

1 Introduction

A positive sequence $(b_n)$ is said to be almost increasing if there exists a positive increasing sequence $c_n$ and two positive constants A and B such that $A{c}_n\le b_n\le B{c}_n$ (see [1]). A sequence $({\lambda }_n)$ is said to be of bounded variation, denoted by $({\lambda }_n)\in \mathcal{BV}$, if ${\sum }_{n=1}^{\mathrm{\infty }}|\mathrm{\Delta }{\lambda }_n|={\sum }_{n=1}^{\mathrm{\infty }}|{\lambda }_n-{\lambda }_{n+1}|<\mathrm{\infty }$. A positive sequence $X=(X_n)$ is said to be a quasi-σ-power increasing sequence if there exists a constant $K=K(\sigma ,X)\ge 1$ such that $K{n}^{\sigma }{X}_n\ge m^{\sigma }{X}_m$ holds for all $n\ge m\ge 1$ (see [2]). It should be noted that every almost increasing sequence is a quasi-σ-power increasing sequence for any nonnegative σ, but the converse may not be true as can be seen by taking an example, say $X_n=n^{-\sigma }$ for $\sigma >0$. Let $({\phi }_n)$ be a sequence of complex numbers and let $\sum a_n$ be a given infinite series with partial sums $(s_n)$. We denote by $z_n^{\alpha }$ and $t_n^{\alpha }$ the n th Cesàro means of order α, with $\alpha >-1$, of the sequences $(s_n)$ and $(n{a}_n)$, respectively, that is,

(1)

(2)

where

$$A_n^{\alpha }=\left(\begin{array}{c}n+\alpha \\ n\end{array}\right)=\frac{(\alpha +1)(\alpha +2)\cdots (\alpha +n)}{n!}=O\left(n^{\alpha }\right),A_{-n}^{\alpha }=0\text{for }n>0.$$

(3)

The series $\sum a_n$ is said to be summable $\phi -{|C,\alpha |}_k$, $k\ge 1$ and $\alpha >-1$, if (see [3, 4])

$$\sum _{n=1}^{\mathrm{\infty }}{\left|{\phi }_n(z_n^{\alpha }-z_{n-1}^{\alpha })\right|}^k=\sum _{n=1}^{\mathrm{\infty }}{n}^{-k}{\left|{\phi }_n{t}_n^{\alpha }\right|}^k<\mathrm{\infty }.$$

(4)

In the special case if we take ${\phi }_n=n^{1-\frac{1}{k}}$, then $\phi -{|C,\alpha |}_k$ summability is the same as ${|C,\alpha |}_k$ summability (see [5]). Also, if we take ${\phi }_n=n^{\delta +1-\frac{1}{k}}$, then $\phi -{|C,\alpha |}_k$ summability reduces to ${|C,\alpha ;\delta |}_k$ summability (see [6]).

2 The known results

Theorem A ([7])

Let $({\lambda }_n)\in \mathcal{BV}$ and let $(X_n)$ be a quasi-σ-power increasing sequence for some σ ($0<\sigma <1$). Suppose also that there exist sequences $({\beta }_n)$ and $({\lambda }_n)$ such that

(5)

(6)

(7)

(8)

If there exists an $ϵ>0$ such that the sequence $(n^{ϵ-k}{|{\phi }_n|}^k)$ is nonincreasing and if the sequence $(w_n^{\alpha })$ defined by (see [8])

$$w_n^{\alpha }=\{\begin{array}{cc}|t_n^{\alpha }|,\hfill & \alpha =1,\hfill \\ {max}_{1\le v\le n}|t_v^{\alpha }|,\hfill & 0<\alpha <1,\hfill \end{array}$$

(9)

satisfies the condition

$$\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k}=O(X_m)\mathit{\text{as}}m\to \mathrm{\infty },$$

(10)

then the series $\sum a_n{\lambda }_n$ is summable $\phi -{|C,\alpha |}_k$, $k\ge 1$, $0<\alpha \le 1$ and k$\alpha +ϵ>1$.

Remark 1 Here, in the hypothesis of Theorem A, we have added the condition ‘$({\lambda }_n)\in \mathcal{BV}$’ because it is necessary.

Theorem B ([9])

Let $(X_n)$ be a quasi-σ-power increasing sequence for some σ ($0<\sigma <1$). If there exists an $ϵ>0$ such that the sequence $(n^{ϵ-k}{|{\phi }_n|}^k)$ is nonincreasing and if the conditions from (5) to (8) are satisfied and if the condition

$$\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}=O(X_m)\mathit{\text{as}}m\to \mathrm{\infty },$$

(11)

is satisfied, then the series $\sum a_n{\lambda }_n$ is summable $\phi -{|C,\alpha |}_k$, $k\ge 1$, $0<\alpha \le 1$ and $k(\alpha -1)+ϵ>1$.

Remark 2 It should be noted that condition (11) is the same as condition (10) when $k=1$. When $k>1$, condition (11) is weaker than condition (10) but the converse is not true. As in [10], we can show that if (10) is satisfied, then we get

$$\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}=O\left(\frac{1}{X_1^{k-1}}\right)\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k}=O(X_m).$$

If (11) is satisfied, then for $k>1$ we obtain that

$$\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k}=\sum _{n=1}^m{X}_n^{k-1}\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}=O\left(X_m^{k-1}\right)\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}=O\left(X_m^k\right)\ne O(X_m).$$

Also, it should be noted that the condition ‘$({\lambda }_n)\in \mathcal{BV}$’ has been removed.

3 The main result

The aim of this paper is to extend Theorem B by using a general class of quasi power increasing sequence instead of a quasi-σ-power increasing sequences. For this purpose, we need the concept of quasi-f-power increasing sequence. A positive sequence $X=(X_n)$ is said to be a quasi-f-power increasing sequence, if there exists a constant $K=K(X,f)$ such that $K{f}_n{X}_n\ge f_m{X}_m$, holds for $n\ge m\ge 1$, where $f=(f_n)=[n^{\sigma }{(logn)}^{\eta },\eta \ge 0,0<\sigma <1]$ (see [11]). It should be noted that if we take $\eta =0$, then we get a quasi-σ-power increasing sequence. Now, we will prove the following theorem.

Theorem Let $(X_n)$ be a quasi-f-power increasing sequence. If there exists an $ϵ>0$ such that the sequence $(n^{ϵ-k}{|{\phi }_n|}^k)$ is non-increasing and if the conditions from (5) to (8) and (11) are satisfied, then the series $\sum a_n{\lambda }_n$ is summable $\phi -{|C,\alpha |}_k$, $k\ge 1$, $0<\alpha \le 1$ and $k(\alpha -1)+ϵ>1$.

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

Lemma 1 ([12])

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

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

(12)

Lemma 2 ([11])

Under the conditions on $(X_n)$, $({\beta }_n)$, and $({\lambda }_n)$ as expressed in the statement of the theorem, we have the following:

(13)

(14)

4 Proof of the theorem

Let $(T_n^{\alpha })$ be the n th $(C,\alpha )$, with $0<\alpha \le 1$, mean of the sequence $(n{a}_n{\lambda }_n)$. Then, by (2), we have

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

(15)

First, applying Abel’s transformation and then using Lemma 1, we get that

To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{-k}{\left|{\phi }_n{T}_{n,r}^{\alpha }\right|}^k<\mathrm{\infty }\text{for }r=1,2.$$

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

$$\begin{array}{rcl}\sum _{n=2}^{m+1}{n}^{-k}{\left|{\phi }_n{T}_{n,1}^{\alpha }\right|}^k& \le & \sum _{n=2}^{m+1}{n}^{-k}{\left(A_n^{\alpha }\right)}^{-k}{|{\phi }_n|}^k{\{\sum _{v=1}^{n-1}{A}_v^{\alpha }{w}_v^{\alpha }|\mathrm{\Delta }{\lambda }_v|\}}^k\\ \le & \sum _{n=2}^{m+1}{n}^{-k}{n}^{-\alpha k}{|{\phi }_n|}^k\sum _{v=1}^{n-1}{v}^{\alpha k}{\left(w_v^{\alpha }\right)}^k{|\mathrm{\Delta }{\lambda }_v|}^k\times {\left\{\sum _{v=1}^{n-1}1\right\}}^{k-1}\\ =& O(1)\sum _{v=1}^m{v}^{\alpha k}{\left(w_v^{\alpha }\right)}^k{({\beta }_v)}^k\sum _{n=v+1}^{m+1}\frac{n^{ϵ-k}{|{\phi }_n|}^k}{n^{k(\alpha -1)+ϵ+1}}\\ =& O(1)\sum _{v=1}^m{v}^{\alpha k}{\left(w_v^{\alpha }\right)}^k{({\beta }_v)}^k{v}^{ϵ-k}{|{\phi }_v|}^k\sum _{n=v+1}^{m+1}\frac{1}{n^{k(\alpha -1)+ϵ+1}}\\ =& O(1)\sum _{v=1}^m{v}^{\alpha k}{\left(w_v^{\alpha }\right)}^k{({\beta }_v)}^k{v}^{ϵ-k}{|{\phi }_v|}^k{\int }_v^{\mathrm{\infty }}\frac{dx}{x^{k(\alpha -1)+ϵ+1}}\\ =& O(1)\sum _{v=1}^m{\beta }_v{({\beta }_v)}^{k-1}{(w_v^{\alpha }|{\phi }_v|)}^k\\ =& O(1)\sum _{v=1}^m{\beta }_v{\left(\frac{1}{v{X}_v}\right)}^{k-1}{(w_v^{\alpha }|{\phi }_v|)}^k\\ =& O(1)\sum _{v=1}^{m-1}\mathrm{\Delta }(v{\beta }_v)\sum _{r=1}^v\frac{{(|{\phi }_r|w_r^{\alpha })}^k}{r^k{X}_r^{k-1}}+O(1)m{\beta }_m\sum _{v=1}^m\frac{{(|{\phi }_v|w_v^{\alpha })}^k}{v^k{X}_v^{k-1}}\\ =& O(1)\sum _{v=1}^{m-1}|\mathrm{\Delta }(v{\beta }_v)|X_v+O(1)m{\beta }_m{X}_m\\ =& O(1)\sum _{v=1}^{m-1}|(v+1)\mathrm{\Delta }{\beta }_v-{\beta }_v|X_v+O(1)m{\beta }_m{X}_m\\ =& O(1)\sum _{v=1}^{m-1}v|\mathrm{\Delta }{\beta }_v|X_v+O(1)\sum _{v=1}^{m-1}{\beta }_v{X}_v+O(1)m{\beta }_m{X}_m\\ =& O(1)\text{as }m\to \mathrm{\infty },\end{array}$$

by virtue of the hypotheses of the theorem and Lemma 2. Finally, we have that

$$\begin{array}{rcl}\sum _{n=1}^m{n}^{-k}{\left|{\phi }_n{T}_{n,2}^{\alpha }\right|}^k& =& \sum _{n=1}^m|{\lambda }_n|{|{\lambda }_n|}^{k-1}{n}^{-k}{(w_n^{\alpha }|{\phi }_n|)}^k\\ =& O(1)\sum _{n=1}^m|{\lambda }_n|{\left(\frac{1}{X_n}\right)}^{k-1}{n}^{-k}{(w_n^{\alpha }|{\phi }_n|)}^k\\ =& O(1)\sum _{n=1}^{m-1}\mathrm{\Delta }|{\lambda }_n|\sum _{v=1}^n\frac{{(|{\phi }_v|w_v^{\alpha })}^k}{v^k{X}_v^{k-1}}+O(1)|{\lambda }_m|\sum _{n=1}^m\frac{{(|{\phi }_n|w_n^{\alpha })}^k}{n^k{X}_n^{k-1}}\\ =& O(1)\sum _{n=1}^{m-1}|\mathrm{\Delta }{\lambda }_n|X_n+O(1)|{\lambda }_m|X_m\\ =& O(1)\sum _{n=1}^{m-1}{\beta }_n{X}_n+O(1)|{\lambda }_m|X_m=O(1)\text{as }m\to \mathrm{\infty },\end{array}$$

by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. If we take $ϵ=1$ and ${\phi }_n=n^{1-\frac{1}{k}}$ (resp. $ϵ=1$, $\alpha =1$ and ${\phi }_n=n^{1-\frac{1}{k}}$), then we get a new result dealing with ${|C,\alpha |}_k$ (resp. ${|C,1|}_k$) summability factors of infinite series. Also, if we take $ϵ=1$ and ${\phi }_n=n^{\delta +1-\frac{1}{k}}$, then we get another new result concerning the ${|C,\alpha ;\delta |}_k$ summability factors of infinite series. Furthermore, if we take $(X_n)$ as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see [13]). Finally, if we take $\eta =0$, then we obtain Theorem B.