Abstract
In the present paper, we obtained a main theorem related to factored infinite series. Some new results are also deduced.
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1 Introduction
Let \(\sum a_{n}\) be a given infinite series with \((s_{n})\) as the sequence of partial sums. In [1], Borwein introduced the \((C,\alpha ,\beta)\) methods in the following form: Let \(\alpha+\beta\ne -1,-2,\ldots\) . Then the \((C,\alpha,\beta)\) mean is defined by
where
The series \(\sum{a_{n}}\) is said to be summable \({ \vert {C},\alpha,\beta ,\sigma;\delta \vert }_{k}\), \(k\geq1\), \(\delta\geq0\), \(\alpha+\beta>-1\), and \(\sigma\in {R}\), if (see [2])
where \({t_{n}^{\alpha,\beta}}\) is the \((C,\alpha,\beta)\) transform of the sequence \((na_{n})\). It should be noted that, for \({\beta=0}\), the \({ \vert {C},\alpha,\beta ,\sigma;\delta \vert }_{k}\) summability method reduces to the \({ \vert {C},\alpha,\sigma;\delta \vert }_{k}\) summability method (see [3]). Let us consider the sequence \((\theta_{n}^{\alpha,\beta})\) which is defined by (see [4])
2 The main result
Here, we shall prove the following theorem.
Theorem
If \((\lambda_{n})\) is a convex sequence (see [5]) such that the series \(\sum\frac{\lambda_{n}}{n}\) is convergent and let \((\theta _{n}^{\alpha,\beta})\) be a sequence defined as in (4). If the condition
holds, then the series \(\sum a_{n} \lambda_{n} \) is summable \({ \vert {C},\alpha,\beta,\sigma;\delta \vert }_{k}\), \(k\geq1\), \(0\leq \delta<\alpha\leq1\), \(\sigma\in{R}\), and \(({\alpha+\beta +1})k-{\sigma(\delta k+k-1)}>1 \).
One should note that, if we set \(\sigma=1\), then we obtain a well-known result of Bor (see [6]).
We will use the following lemmas for the proof of the theorem given above.
Lemma 1
[4]
If \(0<\alpha\leq1\), \(\beta>-1\), and \(1 \leq v \leq n\), then
Lemma 2
[7]
If \((\lambda_{n})\) is a convex sequence such that the series \(\sum\frac{\lambda_{n}}{n}\) is convergent, then \(n{\Delta\lambda_{n}}\rightarrow0\textit{ as }n\rightarrow\infty\) and \(\sum_{n=1}^{\infty} (n+1)\Delta^{2} {\lambda_{n}}\) is convergent.
3 Proof of the theorem
Let \((T_{n}^{\alpha,\beta})\) be the nth \((C,\alpha,\beta)\) mean of the sequence \((n{a_{n}}{\lambda_{n}})\). Then, by (1), we have
First applying Abel’s transformation and then using Lemma 1, we have
In order to complete the proof of the theorem by using Minkowski’s inequality, it is sufficient to show that
For \(k>1\), we can apply Hölder’s inequality with indices k and \({k'}\), where \(\frac{1}{k}+\frac{1}{k'}=1\), and we obtain
by virtue of hypotheses of the theorem and Lemma 2. Similarly, we have
in view of hypotheses of the theorem and Lemma 2. This completes the proof of the theorem.
4 Conclusions
By selecting proper values for α, β, δ, and σ, we have some new results concerning the \({ \vert {C,1} \vert }_{k}\), \({ \vert {C},\alpha \vert }_{k}\), and \({ \vert {C},\alpha ;\delta \vert }_{k}\) summability methods.
References
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Bor, H: On the generalized absolute Cesàro summability. Pac. J. Appl. Math. 2, 217-222 (2010)
Tuncer, AN: On generalized absolute Cesàro summability factors. Ann. Pol. Math. 78, 25-29 (2002)
Bor, H: On a new application of power increasing sequences. Proc. Est. Acad. Sci. 57, 205-209 (2008)
Zygmund, A: Trigonometric Series. Inst. Mat. Polskiej Akademi Nauk, Warsaw (1935)
Bor, H: A new application of convex sequences. J. Class. Anal. 1, 31-34 (2012)
Chow, HC: On the summability factors of Fourier series. J. Lond. Math. Soc. 16, 215-220 (1941)
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Bor, H. Some new results on convex sequences. J Inequal Appl 2017, 165 (2017). https://doi.org/10.1186/s13660-017-1438-4
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DOI: https://doi.org/10.1186/s13660-017-1438-4