Absolute φ−|C,α,β;δ|k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varphi- \vert C, \alpha, \beta; \delta\vert _{k}$\end{document} summability of infinite series

In this paper, we established a generalized theorem on a minimal set of sufficient conditions for absolute summability factors by applying a sequence of a wider class (quasi-power increasing sequence) and the absolute Cesàro φ−|C,α,β;δ|k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varphi-\vert C, \alpha, \beta; \delta \vert _{k}$\end{document} summability for an infinite series. We further obtained well-known applications of the above theorem as corollaries, under suitable conditions.


Introduction
Let ∞ n= a n be an infinite series with sequence of partial sums {s n } and the nth sequence to sequence transformation (mean) of {s n } be given by u n s.t. Before discussing ϕ -|C, α, β; δ| k summability, let us introduce some well-known basic summabilities which are helpful in understanding the ϕ -|C, α, β; δ| k summability.
Definition  The series ∞ n= a n is said to be absolute summable, if lim n→∞ u n = s () and ∞ n= |u nu n- | < ∞.
Definition  ([]) Let t n represent the nth (C, ) means of the sequence (na n ), then the series ∞ n= a n is said to be |C, | k summable for k ≥ , if where then the series ∞ n= a n is said to be ϕ -|C, α, β| k summable.
Definition  For the following condition: the series ∞ n= a n is said to be ϕ -|C, α, β; δ| k summable, where k ≥ , δ ≥  and (ϕ n ) is a sequence of positive real numbers.
Bor gave a number of theorems on absolute summability. In , Bor found the sufficient conditions for an infinite series to be |C, α| k summable [] and |C, α; δ| k summable []. In , he generalized his previous results for |C, α, β| k summability [] and |C, α, β; δ| k summability [], respectively. In , Bor [] generalized the |C, α| k summability factor to the |C, α, β; δ| k summability of an infinite series and in [], he discussed a general class of power increasing sequences and absolute Riesz summability factors of an infinite series. In [], Bor applied |C, α, γ ; β| k summability to obtain the sufficient conditions for an infinite series to be absolute summable. Bor [] gave a new application of quasi-power increasing sequence by applying absolute Cesáro ϕ -|C, α| k summability for an infinity series. Özarslan [] generalized the result on ϕ -|C, | k by a more general absolute ϕ -|C, α| k summability. In , Sonker and Munjal [] determined a theorem on generalized absolute Cesáro summability with the sufficient conditions for an infinite series and in [], they used the concept of triangle matrices for obtaining the minimal set of sufficient conditions of an infinite series to be bounded.

Known results
By using |C, α| k summability, Bor [] gave a minimal set of sufficient conditions for an infinite series to be absolute summable.
Theorem . Let X n be a quasi-f -power increasing sequence for some η ( < η < ). Suppose also that there exists a sequence of numbers (A n ) such that it is ξ -quasi-monotone satisfying the following: A n X n is convergent for all n.
If the conditions are satisfied, then the series a n λ n is |C, α| k summable,  < α ≤  and k ≥ .

Main results
. If we set ζ = , then we get a quasi-η-power increasing sequence [].
With the help of generalized Cesáro ϕ -|C, α, β; δ| k summability, we modernized the results of Bor [] and established the following theorem.
Theorem . Let X n be a quasi-f -power increasing sequence for some η ( < η < ). Suppose also that there exists a ξ -quasi-monotone sequence of numbers (A n ) such that A n X n is convergent for all n.

Lemmas
We need the following lemmas for the proof of our theorem.
is a ξ -quasi-monotone sequence with A n ≤ ξ n and nξ n X n < ∞, then Applying Abel's transformation and Lemma ., we have We use Minkowski's inequality, In order to complete the proof of the theorem, it is sufficient to show that  Hence the proof of the theorem is completed.

Corollaries
Corollary . Let X n be a quasi-f -power increasing sequence for some η ( < η < ) and there exists a sequence of numbers (A n ) such that it is ξ -quasi-monotone satisfying ()-() and the following condition: then the series a n λ n is |C, α, β; δ| k summable, n is given by ().
Proof On putting ϕ n = n in Theorem ., we will get () and the following condition: Here, condition () always holds. We omit the details as the proof is similar to that of Theorem . using the conditions () and () instead of () and ().
Proof On putting δ =  in Theorem ., we will get () and (). We omit the details as the proof is similar to that of Theorem . using the conditions () and () instead of () and ().

Corollary . ([]
) Let X n be a quasi-f -power increasing sequence for some η ( < η < ) and there exists a sequence of numbers (A n ) such that it is ξ -quasi-monotone satisfying ()-() and the following conditions: then the series a n λ n is |C, α| k summable,  < α ≤ , k ≥ , where w α n is given by max ≤v≤n |t α v |,  < α < .

()
Proof On putting ϕ n = n, δ =  and β =  in Theorem ., we will get () and the following condition: Here, condition () always holds. We omit the details as the proof is similar to that of Theorem . using the conditions () and () instead of () and ().

Conclusion
The aim of our paper is to obtain the minimal set of sufficient conditions for an infinite series to be absolute Cesáro ϕ -|C, α, β; δ| k summable. Through the investigation, we may conclude that our theorem is a generalized version which can be reduced for several wellknown summabilities as shown in the corollaries. Further, our theorem has been validated through Corollary ., which is a result of Bor [].