A new application of quasimonotone sequences
We prove a general theorem dealing with generalized absolute Ces`aro summability factors of infinite series. This theorem also includes some new and known results.
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Institute of Mathematics, NAS of Ukraine
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| author | Bor, H. Бор, Х. |
| author_facet | Bor, H. Бор, Х. |
| author_sort | Bor, H. |
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| description | We prove a general theorem dealing with generalized absolute Ces`aro summability factors of infinite series. This theorem also includes some new and known results. |
| first_indexed | 2026-03-24T02:14:48Z |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.5
H. Bor (P. O. Box 121, 06-TR Bahçelievler, Ankara, Turkey)
A NEW APPLICATION OF QUASIMONOTONE SEQUENCES
НОВЕ ЗАСТОСУВАННЯ КВАЗIМОНОТОННИХ ПОСЛIДОВНОСТЕЙ
We prove a general theorem dealing with generalized absolute Cesàro summability factors of infinite series. This theorem
also includes some new and known results.
Доведено загальну теорему про узагальненi абсолютнi фактори сумовностi Чезаро для нескiнченних рядiв. Ця
теорема також включає ряд нових та вiдомих результатiв.
1. Introduction. A positive sequence (bn) is said to be an almost increasing sequence if there exists a
positive increasing sequence (cn) and two positive constants M and N such that Mcn \leq bn \leq Ncn
(see [1]). A sequence (dn) is said to be \delta -quasimonotone, if dn \rightarrow 0, dn > 0 ultimately, and
\Delta dn \geq - \delta n, where \Delta dn = dn - dn+1 and \delta = (\delta n) is a sequence of positive numbers (see [2]).
Let
\sum
an be a given infinite series. We denote by t\alpha ,\beta n the nth Cesàro mean of order (\alpha , \beta ), with
\alpha + \beta > - 1, of the sequence (nan), that is (see [8])
t\alpha ,\beta n =
1
A\alpha +\beta
n
n\sum
v=1
A\alpha - 1
n - vA
\beta
vvav, (1)
where
A\alpha
n =
\Biggl(
n+ \alpha
n
\Biggr)
=
(\alpha + 1)(\alpha + 2) . . . (\alpha + n)
n!
= O(n\alpha ). (2)
Let (\theta \alpha ,\beta n ) be a sequence defined by (see [5])
\theta \alpha ,\beta n =
\left\{
\bigm| \bigm| \bigm| t\alpha ,\beta n
\bigm| \bigm| \bigm| , \alpha = 1, \beta > - 1,
\mathrm{m}\mathrm{a}\mathrm{x}1\leq v\leq n
\bigm| \bigm| \bigm| t\alpha ,\beta v
\bigm| \bigm| \bigm| , 0 < \alpha < 1, \beta > - 1.
(3)
The series
\sum
an is said to be summable | C,\alpha , \beta | k, k \geq 1, if (see [9])
\infty \sum
n=1
1
n
\bigm| \bigm| t\alpha ,\beta n
\bigm| \bigm| k < \infty . (4)
If we take \beta = 0, then | C,\alpha , \beta | k summability reduces to | C,\alpha | k summability (see [10]). Also, if we
take \beta = 0 and \alpha = 1, then we have | C, 1| k summability. In [6], we proved the following theorem
dealing with | C,\alpha , \beta | k summability factors of infinite series.
c\bigcirc H. BOR, 2016
1004 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
A NEW APPLICATION OF QUASIMONOTONE SEQUENCES 1005
Theorem A. Let (\theta \alpha ,\beta n ) be a sequence defined as in (3). Let (Xn) be an almost increasing
sequence such that | \Delta Xn| = O(Xn/n) and \lambda n \rightarrow 0 as n \rightarrow \infty . Suppose that there exists a
sequence of numbers (An) such that it is \delta -quasimonotone with
\sum
n\delta nXn < \infty ,
\sum
AnXn is
convergent, and | \Delta \lambda n| \leq | An| for all n. If the condition
m\sum
n=1
(\theta \alpha ,\beta n )k
n
= O(Xm) as m \rightarrow \infty (5)
satisfies, then the series
\sum
an\lambda n is summable | C,\alpha , \beta | k, 0 < \alpha \leq 1, \beta > - 1, (\alpha + \beta ) > 0, and
k \geq 1.
2. Main result. The aim of this paper is to prove Theorem A under weaker conditions. Now,
we shall prove the following theorem.
Theorem. Let (\theta \alpha ,\beta n ) be a sequence defined as in (3). Let (Xn) be an almost increasing sequence
such that | \Delta Xn| = O(Xn/n) and \lambda n \rightarrow 0 as n \rightarrow \infty . Suppose that there exists a sequence of
numbers (An) such that it is \delta -quasimonotone with
\sum
n\delta nXn < \infty ,
\sum
AnXn is convergent, and
| \Delta \lambda n| \leq | An| for all n. If the condition
m\sum
n=1
(\theta \alpha ,\beta n )k
nXk - 1
n
= O(Xm) as m \rightarrow \infty (6)
satisfies, then the series
\sum
an\lambda n is summable | C,\alpha , \beta | k, 0 < \alpha \leq 1, \beta > - 1, k \geq 1, and
(\alpha + \beta - 1) > 0 .
Remark. It should be noted that condition (6) is the same as condition (5) when k = 1. When
k > 1 condition (6) is weaker than condition (5), but the converse is not true. In fact, as in [11], if
(5) is satisfied, then we get
m\sum
n=1
(\theta \alpha ,\beta n )k
n Xk - 1
n
= O
\Biggl(
1
Xk - 1
1
\Biggr)
m\sum
n=1
(\theta \alpha ,\beta n )k
n
= O(Xm) as m \rightarrow \infty .
To show that the converse is false when k > 1, similar as in [7], the following example is sufficient.
We can take Xn = n\epsilon , 0 < \epsilon < 1, and then construct a sequence (un) such that
(\theta \alpha ,\beta n )k
nXn
k - 1
= Xn - Xn - 1,
whence
m\sum
n=1
(\theta \alpha ,\beta n )k
nXn
k - 1
= Xm = m\epsilon ,
and so
m\sum
n=1
(\theta \alpha ,\beta n )k
n
=
m\sum
n=1
(Xn - Xn - 1)X
k - 1
n =
m\sum
n=1
(n\epsilon - (n - 1)\epsilon )n\epsilon (k - 1) \geq
\geq \epsilon
m\sum
n=1
(n - 1)\epsilon - 1n\epsilon (k - 1) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
1006 H. BOR
= \epsilon
m\sum
n=1
(n - 1)\epsilon k - 1 \sim m\epsilon k
k
as m \rightarrow \infty .
It follows that
1
Xm
m\sum
n=1
(\theta \alpha ,\beta n )k
n
\rightarrow \infty as m \rightarrow \infty
provided k > 1. This shows that (5) implies (6) but not conversely.
We need the following lemmas for the proof of our theorem.
Lemma 1 [3]. Under the conditions of the theorem, we have
| \lambda n| Xn = O(1) as n \rightarrow \infty . (7)
Lemma 2 [4]. Under the conditions of the theorem, we have
nAnXn = O(1) as n \rightarrow \infty , (8)
\infty \sum
n=1
nXn| \Delta An| < \infty . (9)
Lemma 3 [5]. If 0 < \alpha \leq 1, \beta > - 1, and 1 \leq v \leq n, then\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
v\sum
p=0
A\alpha - 1
n - pA
\beta
pap
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{a}\mathrm{x}
1\leq m\leq v
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
m\sum
p=0
A\alpha - 1
m - pA
\beta
pap
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| . (10)
3. Proof of the theorem. Let (T\alpha ,\beta
n ) be the nth (C,\alpha , \beta ) mean of the sequence (nan\lambda n). Then,
by (1), we obtain
T\alpha ,\beta
n =
1
A\alpha +\beta
n
n\sum
v=1
A\alpha - 1
n - vA
\beta
vvav\lambda v.
First, applying Abel’s transformation and then using Lemma 3, we have
T\alpha ,\beta
n =
1
A\alpha +\beta
n
n - 1\sum
v=1
\Delta \lambda v
v\sum
p=1
A\alpha - 1
n - pA
\beta
ppap +
\lambda n
A\alpha +\beta
n
n\sum
v=1
A\alpha - 1
n - vA
\beta
vvav,
| T\alpha ,\beta
n | \leq 1
A\alpha +\beta
n
n - 1\sum
v=1
| \Delta \lambda v|
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
v\sum
p=1
A\alpha - 1
n - pA
\beta
ppap
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| + | \lambda n|
A\alpha +\beta
n
\bigm| \bigm| \bigm| \bigm| \bigm|
n\sum
v=1
A\alpha - 1
n - vA
\beta
vvav
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
A\alpha +\beta
n
n - 1\sum
v=1
A(\alpha +\beta )
v \theta \alpha ,\beta v | \Delta \lambda v| + | \lambda n| \theta \alpha ,\beta n = T\alpha ,\beta
n,1 + T\alpha ,\beta
n,2 .
To complete the proof of the theorem, by Minkowski’s inequality , it is sufficient to show that
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
A NEW APPLICATION OF QUASIMONOTONE SEQUENCES 1007
\infty \sum
n=1
1
n
| T\alpha ,\beta
n,r | k < \infty for r = 1, 2.
Whenever k > 1, we can apply Hölder’s inequality with indices k and k\prime , where
1
k
+
1
k\prime
= 1, we
get
m+1\sum
n=2
1
n
| T\alpha ,\beta
n,1 |
k \leq
m+1\sum
n=2
1
n
\bigm| \bigm| \bigm| \bigm| \bigm| 1
A\alpha +\beta
n
n - 1\sum
v=1
A(\alpha +\beta )
v \theta \alpha ,\beta v \Delta \lambda v
\bigm| \bigm| \bigm| \bigm| \bigm|
k
=
= O(1)
m+1\sum
n=2
1
n1+(\alpha +\beta )k
\Biggl\{
n - 1\sum
v=1
v(\alpha +\beta )k| Av| k(\theta \alpha ,\beta v )k
\Biggr\}
\times
\Biggl\{
n - 1\sum
v=1
1
\Biggr\} k - 1
=
= O(1)
m\sum
v=1
v(\alpha +\beta )k| Av| k(\theta \alpha ,\beta v )k
m+1\sum
n=v+1
1
n2+(\alpha +\beta - 1)k
=
= O(1)
m\sum
v=1
v(\alpha +\beta )k| Av| | Av| k - 1(\theta \alpha ,\beta v )k
\infty \int
v
dx
x2+(\alpha +\beta - 1)k
=
= O(1)
m\sum
v=1
| Av| vk - 1 (\theta \alpha ,\beta v )k
vk - 1Xk - 1
v
= O(1)
m\sum
v=1
v| Av|
(\theta \alpha ,\beta v )k
vXk - 1
v
=
= O(1)
m - 1\sum
v=1
\Delta (v| Av| )
v\sum
p=1
(\theta \alpha ,\beta p )k
pXk - 1
p
+O(1)m| Am|
m\sum
v=1
(\theta \alpha ,\beta v )k
vXk - 1
v
=
= O(1)
m - 1\sum
v=1
| (v + 1)\Delta | Av| - | Av| | Xv +O(1)m| Am| Xm =
= O(1)
m - 1\sum
v=1
v| \Delta Av| Xv +O(1)
m - 1\sum
v=1
| Av| Xv +O(1)m| Am| Xm =
= O(1) as m \rightarrow \infty ,
in view of hypotheses of the theorem and Lemma 2. Again, we obtain
m\sum
n=1
1
n
| T\alpha ,\beta
n,2 |
k =
m\sum
n=1
| \lambda n| | \lambda n| k - 1 (\theta
\alpha ,\beta
n )k
n
= O(1)
m\sum
n=1
| \lambda n|
(\theta \alpha ,\beta n )k
nXk - 1
n
=
= O(1)
m - 1\sum
n=1
| \Delta \lambda n|
n\sum
v=1
(\theta \alpha ,\beta v )k
vXk - 1
v
+O(1)| \lambda m|
m\sum
n=1
(\theta \alpha ,\beta n )k
nXk - 1
n
=
= O(1)
m - 1\sum
v=n
| \Delta \lambda n| Xn +O(1)| \lambda m| Xm =
= O(1)
m - 1\sum
n=1
| An| Xn +O(1)| \lambda m| Xm = O(1) as m \rightarrow \infty ,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
1008 H. BOR
by virtue of the hypotheses of the theorem and Lemma 1.
The theorem is proved.
If we take \beta = 0, then we get a new result dealing with the | C,\alpha | k summability factors. Also if
we take \beta = 0 and \alpha = 1, then we obtain a new result concerning the | C, 1| k summability factors.
References
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Mat. Obshch. – 1956. – 5. – P. 483 – 522 (in Russian).
2. Boas R. P. Quasi positive sequences and trigonometric series // Proc. London Math. Soc. A. – 1965. – 14. – P. 38 – 46.
3. Bor H. An application of almost increasing and \delta -quasi-monotone sequences // J. Inequal. Pure and Appl. Math. –
2000. – 1. – Article 18. – 6 p.
4. Bor H. Corrigendum on the paper in [3] // J. Inequal. Pure and Appl. Math. – 2002. – 3. – Article 16. – 2 p.
5. Bor H. On a new application of quasi power increasing sequences // Proc. Est. Acad. Sci. – 2008. – 57. – P. 205 – 209.
6. Bor H. On generalized absolute Cesàro summability // An. şti. Univ. Iaşi. Mat. (New Ser.). – 2011. – 57. – P. 323 – 328.
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Article ID 793548. – 6 p.
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Math. Soc. – 1957. – 7. – P. 113 – 141.
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P. 2645 – 2648.
Received 30.05.14,
after revision — 19.04.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
|
| id | umjimathkievua-article-1897 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:48Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1f/ae9939c3897b1acdf1c05a4f50abc51f.pdf |
| spelling | umjimathkievua-article-18972019-12-05T09:30:55Z A new application of quasimonotone sequences Нове застосування квазiмонотонних послiдовностей Bor, H. Бор, Х. We prove a general theorem dealing with generalized absolute Ces`aro summability factors of infinite series. This theorem also includes some new and known results. Доведено загальну теорему про узагальненi абсолютнi фактори сумовностi Чезаро для нескiнченних рядiв. Ця теорема також включає ряд нових та вiдомих результатiв. Institute of Mathematics, NAS of Ukraine 2016-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1897 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 7 (2016); 1004-1008 Український математичний журнал; Том 68 № 7 (2016); 1004-1008 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1897/879 Copyright (c) 2016 Bor H. |
| spellingShingle | Bor, H. Бор, Х. A new application of quasimonotone sequences |
| title | A new application of quasimonotone sequences |
| title_alt | Нове застосування квазiмонотонних послiдовностей |
| title_full | A new application of quasimonotone sequences |
| title_fullStr | A new application of quasimonotone sequences |
| title_full_unstemmed | A new application of quasimonotone sequences |
| title_short | A new application of quasimonotone sequences |
| title_sort | new application of quasimonotone sequences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1897 |
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