Article
KYUNGPOOK Math. J. 2019; 59(2): 233-240
Published online June 23, 2019
Copyright © Kyungpook Mathematical Journal.
An Application of Absolute Matrix Summability using Almost Increasing and δ-quasi-monotone Sequences
Hikmet Seyhan Özarslan
Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey
e-mail : seyhan@erciyes.edu.tr and hseyhan38@gmail.com
Received: October 31, 2017; Revised: August 9, 2018; Accepted: August 13, 2018
Abstract
In the present paper, absolute matrix summability of infinite series is studied. A new theorem concerning absolute matrix summability factors, which generalizes a known theorem dealing with absolute Riesz summability factors of infinite series, is proved using almost increasing and
Keywords: summability factors, almost increasing sequences, absolute matrix summability, quasi-monotone sequences, infinite series, Ho¨lder inequality, Minkowski inequality.
1. Introduction
A positive sequence (
Let (
The sequence-to-sequence transformation
defines the sequence (
where
Let
The series ∑
where
When we take
Let
and
and
2. Known Result
In [4, 5], the following theorem dealing with |
Theorem 2.1
3. Main Result
The aim of this paper is to prove following more general theorem dealing with |
Theorem 3.1
Lemma 3.2.([4])
Lemma 3.3.([5])
4. Proof of Theorem 3.1
Let (
Applying Abel’s transformation to above sum, we get
To prove Theorem 3.1, we will show that
First, by using (
Now, as in
Since
by (
by using (
Now, using (
then
by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2. Also, we have
By (
Thus, using (
then we get
by virtue of the hypotheses of Theorem 3.1 and Lemma 3.3. Again, operating Hölder’s inequality, we have
by (
If we take
References
- NK. Bari, and SB. Stečkin.
Best approximations and differential proprerties of two conjugate functions . Tr Mosk Mat Obs.,5 (1956), 483-522. - RP. Boas.
Quasi-positive sequences and trigonometric series . Proc London Math Soc.,14a (Array), 38-46. - H. Bor.
On two summability methods . Math Proc Cambridge Philos Soc.,97 (1985), 147-149. - H. Bor.
An application of almost increasing and δ-quasi-monotone sequences . JIPAM J Inequal Pure Appl Math.,1 (2)(2000) Article 18, 6 pp. - H. Bor.
Corrigendum on the paper “An application of almost increasing and δ-quasimonotone sequences” . JIPAM J Inequal Pure Appl Math.,3 (1)(2002) Article 16, 2 pp. - TM. Flett.
On an extension of absolute summability and some theorems of Littlewood and Paley . Proc London Math Soc.,7 (1957), 113-141. - GH. Hardy. Divergent Series,
, Oxford University Press, Oxford, 1949. - E. Kogbetliantz.
Sur les séries absolument sommables par la méthode des moyennes arithmétiques . Bull Sci Math.,49 (1925), 234-256. - WT. Sulaiman.
Inclusion theorems for absolute matrix summability methods of an infinite series. IV . Indian J Pure Appl Math.,34 (11)(2003), 1547-1557.