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A generalization of a theorem of Bor
Journal of Inequalities and Applications volume 2017, Article number: 179 (2017)
Abstract
In this paper, a general theorem concerning absolute matrix summability is established by applying the concepts of almost increasing and δ-quasi-monotone sequences.
1 Introduction
A positive sequence \((y_{n})\) is said to be almost increasing if there is a positive increasing sequence \((u_{n})\) and two positive constants K and M such that \(Ku_{n} \leq y_{n} \leq Mu_{n}\) (see [1]). A sequence \((c_{n})\) is said to be δ-quasi-monotone, if \(c_{n}\rightarrow 0\), \(c_{n}>0\) ultimately and \(\Delta c_{n}\geq-\delta_{n}\), where \(\Delta c_{n}=c_{n}-c_{n+1}\) and \(\delta=(\delta_{n})\) is a sequence of positive numbers (see [2]). Let \(\sum a_{n}\) be a given infinite series with partial sums \((s_{n})\). Let \(T=(t_{nv})\) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. At that time T describes the sequence-to-sequence transformation, mapping the sequence \(s=(s_{n})\) to \(Ts=(T_{n}(s))\), where
Let \((\varphi_{n})\) be any sequence of positive real numbers. The series \(\sum{a_{n}}\) is said to be summable \({\varphi}-|T,p_{n}|_{k}\), \(k\geq 1\), if (see [3])
where
If we take \(\varphi_{n}=\frac{P_{n}}{p_{n}}\), then \({\varphi }-|T,p_{n}|_{k}\) summability reduces to \(|T,p_{n}|_{k}\) summability (see [4]). If we set \(\varphi_{n}=n\) for all n, \(\varphi- \vert T,p_{n} \vert _{k}\) summability is the same as \(\vert T \vert _{k}\) summability (see [5]). Also, if we take \(\varphi_{n}=\frac{P_{n}}{p_{n}}\) and \(t_{nv}=\frac {p_{v}}{P_{n}}\), then we get \(|\bar{N},p_{n}|_{k}\) summability (see [6]).
2 Known result
In [7, 8], Bor has established the following theorem dealing with \(\vert \bar{N},p_{n} \vert _{k}\) summability factors of infinite series.
Theorem 2.1
Let \((Y_{n})\) be an almost increasing sequence such that \(|\Delta{Y_{n}}|=O(Y_{n}/n)\) and \(\lambda_{n}\rightarrow0\) as \(n\rightarrow\infty\). Assume that there is a sequence of numbers \((B_{n})\) such that it is δ-quasi-monotone with \(\sum nY_{n}\delta_{n}<\infty\), \(\sum B_{n}Y_{n}\) is convergent and \(|\Delta\lambda_{n}|\leq|B_{n}|\) for all n. If
and
where \((z_{n})\) is the nth \((C,1)\) mean of the sequence \((n a_{n})\), then the series \(\sum a_{n}\lambda_{n}\) is summable \(|\bar{N},p_{n}|_{k}\), \(k\geq1\).
3 Main result
The purpose of this paper is to generalize Theorem 2.1 to the \(\varphi - \vert T,p_{n} \vert _{k}\) summability. Before giving main theorem, let us introduce some well-known notations. Let \(T= (t_{nv} )\) be a normal matrix. Lower semimatrices \(\bar{T}=(\bar{t}_{nv})\) and \(\hat{T}=(\hat{t}_{nv})\) are defined as follows:
and
Here, T̄ and T̂ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then we write
and
By taking the definition of general absolute matrix summability, we established the following theorem.
Theorem 3.1
Let \(T=(t_{nv})\) be a positive normal matrix such that
and \((\frac{\varphi_{n}p_{n}}{P_{n}} )\) be a non-increasing sequence. If all conditions of Theorem 2.1 with conditions (4) and (5) are replaced by
and
then the series \(\sum a_{n} \lambda_{n}\) is \({\varphi}-|T,p_{n}|_{k}\) summable, \(k\geq1\).
We need the following lemmas for the proof of Theorem 3.1.
Lemma 3.2
[7]
Let \((Y_{n})\) be an almost increasing sequence and \(\lambda_{n}\rightarrow0\) as \(n\rightarrow\infty\). If \((B_{n})\) is δ-quasi-monotone with \(\sum B_{n}Y_{n}\) is convergent and \(|\Delta\lambda_{n}|\leq|B_{n}|\) for all n, then we have
Lemma 3.3
[8]
Let \((Y_{n})\) be an almost increasing sequence such that \(n| {\Delta Y_{n}}|=O (Y_{n} )\). If \((B_{n})\) is δ-quasi monotone with \(\sum nY_{n}\delta_{n}<\infty\), and \(\sum B_{n}Y_{n}\) is convergent, then
4 Proof of Theorem 3.1
Let \((I_{n})\) indicate the T-transform of the series \(\sum a_{n}\lambda_{n}\). Then we obtain
Using Abel’s formula for (18), we obtain
For the proof of Theorem 3.1, it suffices to prove that
for \(r=1,2,3,4\).
By Hölder’s inequality, we have
Hence, we get
by using (12)
Now, using (11) and (19), we obtain
Thus, by using Abel’s formula, we obtain
Again, using Hölder’s inequality, we have
By means of (6), (7) and (11), we have
In this way, we have
By (6), (7), (10) and (11), we obtain
Thus, using (6) and (10), we have
then we get
in view of (13), (16) and (17).
Also, we have
in view of (3), (12), (13) and (15).
Finally, as in \(I_{n,1}\), we have
in view of (12), (14) and (15). Finally, the proof of Theorem 3.1 is completed.
5 Corollary
If we take \(\varphi_{n}=\frac{P_{n}}{p_{n}}\) and \(t_{nv}=\frac {p_{v}}{P_{n}}\) in Theorem 3.1, then we get Theorem 2.1. In this case, conditions (13) and (14) reduce to conditions (4) and (5), respectively. Also, the condition ‘\((\frac{\varphi_{n}p_{n}}{P_{n}} )\) is a non-increasing sequence’ and the conditions (10)-(12) are clearly satisfied.
6 Conclusions
In this study, we have generalized a well-known theorem dealing with an absolute summability method to a \({\varphi}-|T,p_{n}|_{k}\) summability method of an infinite series by using almost increasing sequences and δ-quasi-monotone sequences.
References
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Acknowledgements
This work was supported by Research Fund of the Erciyes University, Project Number: FBA-2014-3846.
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Özarslan, H.S., Kartal, B. A generalization of a theorem of Bor. J Inequal Appl 2017, 179 (2017). https://doi.org/10.1186/s13660-017-1455-3
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DOI: https://doi.org/10.1186/s13660-017-1455-3