Almost increasing sequences and their new applications

Abstract

In this paper, we generalize a known theorem dealing with $|C,1{|}_k$ summability factors to the ${|C,\alpha |}_k$ summability factors of infinite series using an almost increasing sequence. This theorem also includes some known and new results.

MSC:26D15, 40D15, 40F05, 40G05.

1 Introduction

A positive sequence $(b_n)$ is said to be an almost increasing sequence 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]). Let $\sum a_n$ be a given infinite series with the sequence of partial sums $(s_n)$. By $t_n^{\alpha }$ we denote the n th Cesàro mean of order α, with $\alpha >-1$, of the sequence $(n{a}_n)$, that is,

$$t_n^{\alpha }=\frac{1}{A_n^{\alpha }}\sum _{v=0}^n{A}_{n-v}^{\alpha -1}v{a}_v,$$

(1)

where

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

(2)

The series $\sum a_n$ is said to be summable $|C,\alpha {|}_k$, $k\ge 1$, if (see [2])

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

(3)

If we take $\alpha =1$, then ${|C,\alpha |}_k$ summability reduces to $|C,1{|}_k$ summability.

2 Known result

Many works dealing with an application of almost increasing sequences to the absolute Cesàro summability factors of infinite series have been done (see [3–11]). Among them, in [10], the following main theorem dealing with $|C,1{|}_k$ summability factors has been proved.

Theorem A Let $({\phi }_n)$ be a positive sequence and $(X_n)$ be an almost increasing sequence. If the conditions

$$\sum _{n=1}^{\mathrm{\infty }}n\left|{\mathrm{\Delta }}^2{\lambda }_n\right|X_n<\mathrm{\infty },$$

(4)

$$|{\lambda }_n|X_n=O(1)\mathit{\text{as}}n\to \mathrm{\infty },$$

(5)

$${\phi }_n=O(1)\mathit{\text{as}}n\to \mathrm{\infty },$$

(6)

$$n\mathrm{\Delta }{\phi }_n=O(1)\mathit{\text{as}}n\to \mathrm{\infty },$$

(7)

$$\sum _{v=1}^n\frac{|t_v{|}^k}{v{X}_v^{k-1}}=O(X_n)\mathit{\text{as}}n\to \mathrm{\infty }$$

(8)

are satisfied, then the series $\sum a_n{\lambda }_n{\phi }_n$ is summable $|C,1{|}_k$, $k\ge 1$.

3 The main result

The aim of this paper is to generalize Theorem A to the ${|C,\alpha |}_k$ summability in the following form.

Theorem Let $({\phi }_n)$ be a positive sequence and let $(X_n)$ be an almost increasing sequence.

If the conditions (4), (5), (6) and (7) are satisfied, and the sequence $(w_n^{\alpha })$ defined by (see [12])

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

(9)

satisfies the condition

$$\sum _{v=1}^n\frac{{(w_v^{\alpha })}^k}{v{X}_v^{k-1}}=O(X_n)\mathit{\text{as}}n\to \mathrm{\infty },$$

(10)

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

Remark It should be noted that if we take $\alpha =1$, then we get Theorem A. In this case, condition (10) reduces to condition (8) and the condition ‘$(\alpha -1)k>-1$’ is trivial.

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

Lemma 1 [13]

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

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

(11)

Lemma 2 [14]

Under the conditions (4) and (5), we have

$$n{X}_n|\mathrm{\Delta }{\lambda }_n|=O(1)\mathit{\text{as}}n\to \mathrm{\infty },$$

(12)

$$\sum _{n=1}^{\mathrm{\infty }}{X}_n|\mathrm{\Delta }{\lambda }_n|<\mathrm{\infty }.$$

(13)

4 Proof of the Theorem

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

Then, by (1), we find that

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

(14)

Thus, applying Abel’s transformation first and then using Lemma 1, we have that

$$\begin{array}{c}\begin{array}{rl}{T}_n^{\alpha }& =\frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}\mathrm{\Delta }({\lambda }_v{\phi }_n)\sum _{p=1}^v{A}_{n-p}^{\alpha -1}p{a}_p+\frac{{\lambda }_n{\phi }_n}{A_n^{\alpha }}\sum _{v=1}^n{A}_{n-v}^{\alpha -1}v{a}_v\\ =\frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}({\lambda }_v\mathrm{\Delta }{\phi }_v+{\phi }_{v+1}\mathrm{\Delta }{\lambda }_v)\sum _{p=1}^v{A}_{n-p}^{\alpha -1}p{a}_p+\frac{{\lambda }_n{\phi }_n}{A_n^{\alpha }}\sum _{v=1}^n{A}_{n-v}^{\alpha -1}v{a}_v,\end{array}\hfill \\ \begin{array}{rl}\left|T_n^{\alpha }\right|\le & \frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}|{\lambda }_v\mathrm{\Delta }{\phi }_v|\left|\sum _{p=1}^v{A}_{n-p}^{\alpha -1}p{a}_p\right|+\frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}|{\phi }_{v+1}\mathrm{\Delta }{\lambda }_v|\left|\sum _{p=1}^v{A}_{n-p}^{\alpha -1}p{a}_p\right|\\ +\frac{|{\lambda }_n{\phi }_n|}{A_n^{\alpha }}\left|\sum _{v=1}^v{A}_{n-v}^{\alpha -1}v{a}_v\right|\\ \le & \frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}{A}_v^{\alpha }{w}_v^{\alpha }|{\lambda }_v||\mathrm{\Delta }{\phi }_v|+\frac{1}{A_n^{\alpha }}\sum _{v=1}^{n-1}{A}_v^{\alpha }{w}_v^{\alpha }|{\phi }_{v+1}||\mathrm{\Delta }{\lambda }_v|+|{\lambda }_n||{\phi }_n|w_n^{\alpha }\\ =& T_{n,1}^{\alpha }+T_{n,2}^{\alpha }+T_{n,3}^{\alpha }.\end{array}\hfill \end{array}$$

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

$$\sum _{n=1}^{\mathrm{\infty }}{n}^{-1}{\left|T_{n,r}^{\alpha }\right|}^k<\mathrm{\infty }\text{for }r=1,2,3.$$

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

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

by virtue of the hypotheses of the theorem and Lemma 2. Again, we get that

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

by hypotheses of the theorem and Lemma 2. Finally, as in $T_{n,1}^{\alpha }$, we have that

$$\begin{array}{rcl}\sum _{n=1}^m{n}^{-1}{\left|T_{n,3}^{\alpha }\right|}^k& =& \sum _{n=1}^m{n}^{-1}{\left|{\lambda }_n{\phi }_n{w}_n^{\alpha }\right|}^k\\ =& O(1)\sum _{n=1}^m\frac{{(w_n^{\alpha })}^k|{\lambda }_n|}{n{X}_n^{k-1}}=O(1)\text{as }m\to \mathrm{\infty }\end{array}$$

by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. Also, if we take $k=1$, then we get a new result concerning the $|C,\alpha |$ summability factors of infinite series.