A generalization of a theorem of Bor

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.

A Note on Summability of Infinite Series

“…Let us mention about some of the recent ones. Karakaş [5,6] , Kartal [7,8], Özarslan [9][10][11][12], Özarslan and Kartal [13,14], Özarslan, Şakar and Kartal [15] proved theorems on the sufficient conditions to absolute matrix summability of infinite series by using different class of sequences. The object of this paper is to get the necessary and sufficient conditions to absolute matrix summability.…”

Necessary and sufficient conditions for the inclusion relation between two summability methods

“…Sezer and Çanak [20], Bor [2] obtained equivalence theorems on summability. Özgen [19], Özarslan and Karakaş [14], Özarslan and Kartal [15], [17], Sonker and Munjal [21], [22], Karakaş [7], Kartal [8], [9], Özarslan [10], [11], Bor and Agarwal [3], Bor and Mohapatra [4] obtained theorems on absolute Riesz, Cesàro and matrix summability of infinite series. Furthermore, the following theorem on the relative strength of two absolute matrix summability methods has been proved in [13].…”

“…In this paper, a theorem on the relative strength of two absolute matrix summability methods is generalized. In case of β = 1 and δ = 0, the conditions (15), (16) reduce to the conditions (10), (11) respectively and so Theorem 2 reduces to Theorem 1. Also, in case of β = 1, Theorem 2 reduces to the result in [12].…”

On absolute summability of infinite series and Fourier series

“…There are many papers on the field of summability, some of them are [4,5,8,9,10,12,13,14,15,16]. In this paper, Theorem 1 is generalized to the θ − |T ; δ| k summability method as in the following.…”