A new application of generalized quasi-power increasing sequences
We prove a theorem dealing with $|\overline{N}, p_n, \theta_n|_k$-summability using a new general class of power increasing sequences instead of a quasi-$\eta$-power increasing sequence. This theorem also includes some new and known results.
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2012
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| author | Bor, H. Бор, Х. |
| author_facet | Bor, H. Бор, Х. |
| author_sort | Bor, H. |
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| datestamp_date | 2020-03-18T19:30:57Z |
| description | We prove a theorem dealing with $|\overline{N}, p_n, \theta_n|_k$-summability using a new general class of power increasing sequences instead of a quasi-$\eta$-power increasing sequence. This theorem also includes some new and known results. |
| first_indexed | 2026-03-24T02:26:48Z |
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UDC 517.5
H. Bor (Ankara, Turkey)
A NEW APPLICATION
OF GENERALIZED QUASI-POWER INCREASING SEQUENCES
НОВЕ ЗАСТОСУВАННЯ УЗАГАЛЬНЕНИХ ПОСЛIДОВНОСТЕЙ
КВАЗIСТЕПЕНЕВОГО ЗРОСТАННЯ
We prove a theorem dealing with | N̄ , pn, θn |k-summability using a new general class of power increasing sequences
instead of a quasi-η-power increasing sequence. This theorem also includes some new and known results.
Доведено теорему про | N̄ , pn, θn |k-сумовнiсть iз використанням нового загального класу послiдовностей степе-
невого зростання замiсть послiдовностi квазi-η-степеневого зростання. Окремими випадками цiєї теореми є деякi
новi та вiдомi результати.
1. Introduction. A positive sequence (bn) is said to be almost increasing if there exists a positive
increasing sequence (cn) and two positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]).
We write BVO = BV ∩ CO, where CO = { x = (xk) ∈ Ω: limk |xk| = 0} , BV =
{
x = (xk) ∈ Ω:∑
k
|xk − xk+1| < ∞
}
and Ω being the space of all real-valued sequences. A positive sequence
(δn) is said to be a quasi-η-power increasing sequence if there exists a constant K = K(η, δ) ≥ 1
such that Knηδn ≥ mηδm holds for all n ≥ m ≥ 1 (see [9]). Let
∑
an be a given infinite
series with partial sums (sn). We denote by tn the nth (C,1) mean of the sequence (nan), that is,
tn =
1
n
∑n
v=1
vav. A series
∑
an is said to be summable |C, 1|k, k ≥ 1, if (see [7])
∞∑
n=1
1
n
|tn|k <∞. (1)
Let (pn) be a sequence of positive real numbers such that
Pn =
n∑
v=0
pv →∞ as n→∞ (P−i = p−i = 0, i ≥ 1). (2)
The sequence-to-sequence transformation
σn =
1
Pn
n∑
v=0
pvsv (3)
defines the sequence (σn) of the Riesz mean or simply the (N̄ , pn) mean of the sequence (sn),
generated by the sequence of coefficients (pn) (see [8]). The series
∑
an is said to be summable
|N̄ , pn|k, k ≥ 1, if (see [2])
c© H. BOR, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6 731
732 H. BOR
∞∑
n=1
(Pn/pn)k−1|∆σn−1|k <∞, (4)
where
∆σn−1 = σn − σn−1 = − pn
PnPn−1
n∑
v=1
Pv−1av, n ≥ 1. (5)
In the special case pn = 1 for all values of n, |N̄ , pn|k summability is the same as |C, 1|k summa-
bility. Let (θn) be any sequence of positive constants. The series
∑
an is said to be summable
|N̄ , pn, θn|k, k ≥ 1, if (see [11])
∞∑
n=1
θk−1n |∆σn−1|k <∞. (6)
If we take θn =
Pn
pn
, then |N̄ , pn, θn|k summability reduces to |N̄ , pn|k summability. Also, if we take
θn = n and pn = 1 for all values of n, then we get |C, 1|k summability.
Furthermore, if we take θn = n, then |N̄ , pn, θn|k summability reduces to |R, pn|k (see [4])
summability.
2. Known result. In [6], we have proved the following main theorem dealing with |N̄ , pn, θn|k
summability factors of infinite series.
Theorem A. Let
(
θnpn
Pn
)
be a non-increasing sequence, (λn) ∈ BVO and (Xn) be a quasi-
η-power increasing sequence for some η (0 < η < 1). Suppose also that there exist sequences (βn)
and (λn) such that
|∆λn| ≤ βn, (7)
βn → 0 as n→∞, (8)
∞∑
n=1
n|∆βn|Xn <∞, (9)
|λn|Xn = O(1). (10)
If
n∑
v=1
θk−1v v−k|sv|k = O(Xn) as n→∞, (11)
and (pn) is a sequence such that
Pn = O(npn), (12)
Pn∆pn = O(pnpn+1), (13)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
A NEW APPLICATION OF GENERALIZED QUASI-POWER INCREASING SEQUENCES 733
then the series
∑∞
n=1
an
Pnλn
npn
is summable |N̄ , pn, θn|k, k ≥ 1. If we take (Xn) as an almost
increasing sequence and θn =
Pn
pn
in Theorem A, then we get a result which was published in [3],
in this case the condition “
(
θnpn
Pn
)
is a non-increasing sequence” is automatically satisfied and the
condition (λn) ∈ BVO is not needed.
Remark. It should be noted that, we can take (λn) ∈ BV instead of (λn) ∈ BVO and it is
sufficient to prove Theorem A.
3. Main result. In the present paper, we have generalized Theorem A by using a quasi-f -power
increasing sequence instead of a quasi η-power increasing sequence. For this purpose, we need the
concept of a quasi-f -power increasing sequence. A positive sequence α = (αn) is said to be a quasi-
f -power increasing sequence, if there exists a constantK = K(α, f) ≥ 1 such thatKfnαn ≥ fmαm,
holds for n ≥ m ≥ 1, where f = (fn) =
[
nη(log n)σ, σ ≥ 0, 0 < η < 1
]
(see [12]). It should be
noted that, if we take σ=0, then we get a quasi-η-power increasing sequence.
Now, we shall prove the following general theorem.
Theorem . Let
(
θnpn
Pn
)
be a non-increasing sequence, (λn) ∈ BV and (Xn) be a quasi-f-
power increasing sequence. If the conditions (7) – (13) of Theorem A are satisfied, then the series∑∞
n=1
an
Pnλn
npn
is summable |N̄ , pn, θn|k, k ≥ 1.
If we take σ = 0, then we have Theorem A.
We require the following lemmas for the proof of the theorem.
Lemma 1. Except for the condition (λn) ∈ BV, under the conditions on (Xn), (βn) and (λn)
as expressed in the statement of the theorem, we have the following :
nXnβn = O(1), (14)
∞∑
n=1
βnXn <∞. (15)
Proof. Since βn → 0, then we have ∆βn → 0, and hence
∞∑
n=1
βnXn ≤
∞∑
n=1
Xn
∞∑
v=n
|∆βv| =
∞∑
v=1
|∆βv|
v∑
n=1
Xn =
=
∞∑
v=1
|∆βv|
v∑
n=1
nη(log n)σXnn
−η(log n)−σ =
= O(1)
∞∑
v=1
|∆βv|vη(log v)σXv
v∑
n=1
n−η(log n)−σ =
= O(1)
∞∑
v=1
|∆βv|vη(log v)σXv
v∑
n=1
nε(log n)−σn−η−ε =
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
734 H. BOR
= O(1)
∞∑
v=1
|∆βv|vηXv(log v)σvε(log v)−σ
v∑
n=1
n−η−ε =
= O(1)
∞∑
v=1
|∆βv|vη+εXv
v∫
0
x−η−εdx =
= O(1)
∞∑
v=1
|∆βv|vη+εXvv
1−η−ε =
= O(1)
∞∑
v=1
v|∆βv|Xv = O(1), 0 < ε < η + ε < 1.
Again, we have that
nβnXn = nXn
∞∑
v=n
∆βv ≤ nXn
∞∑
v=n
|∆βv| =
= n1−η(log n)−σnη(log n)σXn
∞∑
v=n
|∆βv| ≤
≤ n1−η(log n)−σ
∞∑
v=n
vη(log v)σXv|∆βv| ≤
≤
∞∑
n=v
v1−η(log v)−σXvv
η(log v)σ|∆βv| =
=
∞∑
v=1
vXv|∆βv| = O(1).
Lemma 1 is proved.
Lemma 2 [10]. If the conditions (12) and (13) are satisfied, then we have that
∆
(
Pn
npn
)
= O
(
1
n
)
. (16)
4. Proof of the theorem. Let (Tn) be the sequence of (N̄ , pn) mean of the series
∑∞
n=1
anPnλn
npn
.
Then, by definition, we have
Tn =
1
Pn
n∑
v=1
pv
v∑
r=1
arPrλr
rpr
=
1
Pn
n∑
v=1
(Pn − Pv−1)
avPvλv
vpv
. (17)
Then
Tn − Tn−1 =
pn
PnPn−1
n∑
v=1
Pv−1Pvavλv
vpv
, n ≥ 1. (18)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
A NEW APPLICATION OF GENERALIZED QUASI-POWER INCREASING SEQUENCES 735
Using Abel’s transformation, we get
Tn − Tn−1 =
pn
PnPn−1
n∑
v=1
sv∆
(
Pv−1Pvλv
vpv
)
+
λnsn
n
=
=
snλn
n
+
pn
PnPn−1
n−1∑
v=1
sv
Pv+1Pv∆λv
(v + 1)pv+1
+
+
pn
PnPn−1
n−1∑
v=1
Pvsvλv∆
(
Pv
vpv
)
− pn
PnPn−1
n−1∑
v=1
svPvλv
1
v
=
= Tn,1 + Tn,2 + Tn,3 + Tn,4, say.
To prove the theorem, by Minkowski’s inequality, it is sufficient to show for k ≥ 1
∞∑
n=1
θk−1n |Tn,r|k <∞ for r = 1, 2, 3, 4. (19)
Firstly, by using Abel’s transformation, we have that
m∑
n=1
θk−1n |Tn,1|k =
m∑
n=1
θk−1n n−k|λn|k−1|λn||sn|k =
= O(1)
m∑
n=1
|λn|θk−1n n−k|sn|k =
= O(1)
m−1∑
n=1
∆|λn|
n∑
v=1
θk−1v v−k|sv|k +O(1)|λm|
m∑
n=1
θk−1n n−k|sn|k =
= O(1)
m−1∑
n=1
|∆λn|Xn +O(1)|λm|Xm =
= O(1)
m−1∑
n=1
βnXn +O(1)|λm|Xm = O(1) as m→∞
by virtue of (7), (10), (11) and (15).
Now, using the fact that Pv+1 = O ((v + 1)pv+1) by (12), and applying Hölder’s inequality we
have that
m+1∑
n=2
θk−1n |Tn,2|k = O(1)
m+1∑
n=2
θk−1n
(
pn
Pn
)k 1
P kn−1
∣∣∣∣∣
n−1∑
v=1
Pvsv∆λv
∣∣∣∣∣
k
=
= O(1)
m+1∑
n=2
θk−1n
(
pn
Pn
)k 1
P kn−1
{
n−1∑
v=1
Pv
pv
|sv|pv|∆λv|
}k
=
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
736 H. BOR
= O(1)
m+1∑
n=2
θk−1n
(
pn
Pn
)k 1
Pn−1
n−1∑
v=1
(
Pv
pv
)k
|sv|kpv (βv)
k
(
1
Pn−1
n−1∑
v=1
pv
)k−1
=
= O(1)
m∑
v=1
(
Pv
pv
)k
|sv|kpv (βv)
k
m+1∑
n=v+1
(
θnpn
Pn
)k−1 pn
PnPn−1
=
= O(1)
m∑
v=1
(
Pv
pv
)k
|sv|kpv (βv)
k
(
θvpv
Pv
)k−1 m+1∑
n=v+1
pn
PnPn−1
=
= O(1)
m∑
v=1
(
Pv
pv
)k
|sv|k (βv)
k
(
pv
Pv
)
θk−1v
(
pv
Pv
)k−1
=
= O(1)
m∑
v=1
(vβv)
k−1vβv
1
vk
θk−1v |sv|k =
= O(1)
m∑
v=1
vβvθ
k−1
v v−k|sv|k =
= O(1)
m−1∑
v=1
∆(vβv)
v∑
r=1
θk−1r r−k|sr|k +O(1)mβm
m∑
v=1
θk−1v v−k|sv|k =
= O(1)
m−1∑
v=1
|∆(vβv)|Xv +O(1)mβmXm =
= O(1)
m−1∑
v=1
v|∆βv|Xv +O(1)
m−1∑
v=1
βvXv +O(1)mβmXm = O(1)
as m→∞, in view of (7), (9), (11), (14) and (15).
Again, we have that
m+1∑
n=2
θk−1n |Tn,3|k = O(1)
m+1∑
n=2
θk−1n
(
pn
Pn
)k 1
P kn−1
{
n−1∑
v=1
Pv|sv||λv|
1
v
}k
=
= O(1)
m+1∑
n=2
θk−1n
(
pn
Pn
)k 1
Pn−1
n−1∑
v=1
(
Pv
pv
)k
v−kpv|sv|k|λv|k
{
1
Pn−1
n−1∑
v=1
pv
}k−1
=
= O(1)
m∑
v=1
(
Pv
pv
)k
v−k|sv|kpv|λv|k
m+1∑
n=v+1
(
θnpn
Pn
)k−1 pn
PnPn−1
=
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
A NEW APPLICATION OF GENERALIZED QUASI-POWER INCREASING SEQUENCES 737
= O(1)
m∑
v=1
(
Pv
pv
)k−1
v−kθk−1v
(
pv
Pv
)k−1
|λv|k−1|λv||sv|k =
= O(1)
m∑
v=1
|λv|θk−1v v−k|sv|k =
= O(1)
m−1∑
v=1
βvXv +O(1)|λm|Xm = O(1) as m→∞,
in view of (7), (10), (11), (15) and (16).
Finally, using Hölder’s inequality, as in Tn,3 we have that
m+1∑
n=2
θk−1n |Tn,4|k =
m+1∑
n=2
θk−1n
(
pn
Pn
)k 1
P kn−1
∣∣∣∣∣
n−1∑
v=1
sv
Pv
v
λv
∣∣∣∣∣
k
=
= O(1)
m+1∑
n=2
θk−1n
(
pn
Pn
)k 1
P kn−1
∣∣∣∣∣
n−1∑
v=1
sv
Pv
vpv
pvλ
∣∣∣∣∣
k
=
= O(1)
m+1∑
n=2
θk−1n
(
pn
Pn
)k 1
Pn−1
n−1∑
v=1
|sv|k
(
Pv
pv
)k
v−kpv|λv|k
(
1
Pn−1
n−1∑
v=1
pv
)k−1
=
= O(1)
m∑
v=1
(
Pv
pv
)k
v−k|sv|kpv|λv|k
1
Pv
(
θvpv
Pv
)k−1
=
= O(1)
m∑
v=1
(
Pv
pv
)k−1
v−k
(
pv
Pv
)k−1
θk−1v |λv|k−1|λv||sv|k =
= O(1)
m∑
v=1
|λv|θk−1v v−k|sv|k =
= O(1)
m−1∑
v=1
βvXv +O(1)|λm|Xm = O(1) as m→∞.
Therefore, we get that
m∑
n=1
θk−1n |Tn,r|k = O(1) as m→∞, for r = 1, 2, 3, 4.
This completes the proof of the theorem. If we take pn = 1 for all values of n, then we have a new
result for |C, 1, θn|k summability. Furthermore, if we take θn = n, then we have another new result
for |R, pn|k summability. Finally, if we take pn = 1 for all values of n and θn = n, then we get a
new result dealing with |C, 1|k summability factors.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
738 H. BOR
1. Bari N. K., Stečkin S. B. est approximation and differential proprerties of two conjugate functions (in Russian) //
Trudy Mosk. Mat. Obshch. – 1956. – 5. – P. 483 – 522.
2. Bor H. On two summability methods // Math. Proc. Cambridge Phil. Soc. – 1985. – 97. – P. 147 – 149.
3. Bor H. A note on |N̄ , pn|k summability factors of infinite series // Indian J. Pure and Appl. Math. – 1987. – 18. –
P. 330 – 336.
4. Bor H. On the relative strength of two absolute summability methods // Proc. Amer. Math. Soc. – 1991. – 113. –
P. 1009 – 1012.
5. Bor H. A general note on increasing sequences // J. Inequal. Pure and Appl. Math. – 2007. – 8. – Article 82
(electronic).
6. Bor H. New application of power increasing sequences // An. şti. Univ. Iaşi. Mat. (N.S.) (to appear).
7. Flett T. M. On an extension of absolute summability and some theorems of Littlewood and Paley // Proc. London
Math. Soc. – 1957. – 7. – P. 113 – 141.
8. Hardy G. H. Divergent series. – Oxford: Oxford Univ. Press, 1949.
9. Leindler L. A new application of quasi power increasing sequences // Publ. Math. Debrecen. – 2001. – 58. –
P. 791 – 796.
10. Mishra K. N., Srivastava R. S. L. On |N̄ , pn| summability factors of infinite series // Indian J. Pure and Appl. Math.
– 1984. – 15. – P. 651 – 656.
11. Sulaiman W. T. On some summability factors of infinite series // Proc. Amer. Math. Soc. – 1992. – 115. – P. 313 – 317.
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– P. 1224 – 1230.
Received 31.08.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
|
| id | umjimathkievua-article-2612 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:48Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6e/acade410adc795699dbd937d3764826e.pdf |
| spelling | umjimathkievua-article-26122020-03-18T19:30:57Z A new application of generalized quasi-power increasing sequences Нове застосування узагальнених послiдовностей квазiстепеневого зростання Bor, H. Бор, Х. We prove a theorem dealing with $|\overline{N}, p_n, \theta_n|_k$-summability using a new general class of power increasing sequences instead of a quasi-$\eta$-power increasing sequence. This theorem also includes some new and known results. Доведено теорему про $|\overline{N}, p_n, \theta_n|_k$-сумовнiсть iз використанням нового загального класу послiдовностей степеневого зростання замiсть послiдовностi квазi-$\eta$-степеневого зростання. Окремими випадками цiєї теореми є деякi новi та вiдомi результати. Institute of Mathematics, NAS of Ukraine 2012-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2612 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 6 (2012); 731-738 Український математичний журнал; Том 64 № 6 (2012); 731-738 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2612/1983 https://umj.imath.kiev.ua/index.php/umj/article/view/2612/1984 Copyright (c) 2012 Bor H. |
| spellingShingle | Bor, H. Бор, Х. A new application of generalized quasi-power increasing sequences |
| title | A new application of generalized quasi-power increasing sequences |
| title_alt | Нове застосування узагальнених послiдовностей квазiстепеневого зростання |
| title_full | A new application of generalized quasi-power increasing sequences |
| title_fullStr | A new application of generalized quasi-power increasing sequences |
| title_full_unstemmed | A new application of generalized quasi-power increasing sequences |
| title_short | A new application of generalized quasi-power increasing sequences |
| title_sort | new application of generalized quasi-power increasing sequences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2612 |
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