ISSN 2291-8639
Volume 15, Number 1 (2017), 108-113
http://www.etamaths.com
A NOTE ON ABSOLUTE CES `ARO ϕ− |C,1;δ;l|k SUMMABILITY FACTOR
SMITA SONKER1, XH. Z. KRASNIQI2,∗AND ALKA MUNJAL1
Abstract. A positive non-decreasing sequence has been used to establish a theorem on a minimal set of sufficient conditions for an infinite series to be absolute Ces`aroϕ− |C,1;δ;l|ksummable. For
some well-known applications, suitable conditions have been applied on the presented theorem for obtaining the sub-result of the presented theorem.
1. Introduction
Let{sn}be a sequence of partial sums of the series
∞
P
n=0
an andnthmean of{sn}is given bytn s.t.
tn =
∞
X
k=0
tnksk (1.1)
where{tnk} is the sequence of the coefficients of the matrix. If sequence of the means{tn} satisfied the following conditions:
lim
n→∞tn=s, (1.2)
and
∞
X
n=1
|tn−tn−1|<∞, (1.3)
then the series ∞
P
n=0
an is said to be absolute summable. If τn represent the nth(C,1) means of the
sequence (nan),then series
∞
P
n=0
an is said to be summable|C,1|k, k≥1 [9], if
∞
X
n=1 1
n|τn|
k <∞. (1.4)
If the sequence{τn} satisfied the condition:
∞
X
n=1
ϕk−1
n
nk |τn| k <
∞, (1.5)
then the series ∞
P
n=0
an is said to be summableϕ− |C,1|k, k≥1,and if the sequence{τn}satisfied the
following condition:
∞
X
n=1
ϕk−1
n
nk−δk|τn|
k<∞, (1.6)
then the series ∞
P
n=0
an isϕ− |C,1;δ|k,summable, wherek≥1, δ≥0 and (ϕn) be a sequence of positive
real numbers.
Received 27th May, 2017; accepted 1stAugust, 2017; published 1stSeptember, 2017. 2010Mathematics Subject Classification. 40F05, 40D15, 40G05.
Key words and phrases. absolute summability; infinite series;ϕ− |C,1;δ;l|ksummability; sequence space.
c
2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
− | |k
Forϕ− |C,1;δ;l|k summability, the infinite series
∞
P
n=0
an satisfied
∞
X
n=1
ϕln(k−1)
nl(k−δk)|τn|
k<∞ (1.7)
wherek≥1, δ≥0 andl is a real number.
Note: If we take l = 1, then ϕ− |C,1;δ;l|k reduces to ϕ− |C,1, δ|k, if ϕ = n, then ϕ− |C,1;δ|k summability reduces to|C,1;δ|k summability and if δ= 0,then|C,1;δ|k reduces to|C,1|k.
In 1972, Mazhar [8] determined the minimal set of sufficient conditions for an infinite series to be absolute |C,1|k summable. This result became an essence of many results found in previous years. In 1980, Balci [10] defined Absoluteϕ-summability factors and determined a very interesting result. Bor gave a number of theorems on absolute summability by a generalization of Mazhar [8] results. In 1993 [2] and 1994 [3], he used|N , p¯ n|k summability for enhancement of the results of Mazhar [8] and apply it on Fourier series. ¨Ozarslan [4] generalized the result of Bor [1] by a more general absolute
ϕ− |C, α|k summability and in [6], he used absolute matrixϕ− |A, δ|k summability and improve some well-known results. Concerning theϕ−|N, pn|ksummability factors, Saxena [7] gave a general theorem for an infinite series.
2. Known results
Absoluteϕ− |C,1|k summability has been used by ¨Ozarslan [5] to establish the following theorem.
Theorem 2.1. Let ϕn be a sequence of positive real numbers, if
λm=O(1), m→ ∞, (2.1)
m
X
n=1
nlogn|∆2λn|=O(1), (2.2)
m
X
v=1
ϕk−1
v
vk |tv|
k =O(logm), as m→ ∞, (2.3)
m
X
n=v
ϕkn−1 nk+1 =O
ϕkv−1 vk
!
. (2.4)
Then the infinite series P
anλn isϕ− |C,1|k summable for k≥1.
3. Main results
Generalized Ces´aroϕ− |C,1;δ;l|k summability and a positive non-decreasing sequence have been used to moderate the conditions of ¨Ozarslan [5] results for an infinite series.
Theorem 3.1. Let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing sequence satisfying the following conditions:
λm=O(1), m→ ∞, (3.1)
m
X
n=1
nlogn|∆2λn|=O(1), (3.2)
m
X
v=1
ϕlv(k−1)
vl(k−δk)|tv|
k =O(logm.µ
m)as m→ ∞, (3.3)
m
X
n=v
ϕln(k−1)
n1+l(k−δk) =O
ϕl(k−1)
v
vl(k−δk)
, (3.4)
nlognµn∆
1 µn
=O(1). (3.5)
Then the infinite seriesPa
4. Proof of the Theorem
LetTn be thenth(C,1) mean of the sequence (nanλn/µn).The series isϕ− |C,1;δ;l|k summable, if
∞
X
n=1
ϕln(k−1)
nl(k−δk)|Tn|
k<∞. (4.1)
Applying Able’s transformation, we have
Tn = 1
n+ 1 n
X
v=1
vavλv
µv
= 1
n+ 1 n−1
X
v=1
v
X
r=1
rar
!
∆ λv
µv
!
+ λv
µv
! n
X
v=1
vav
!
= 1
n+ 1 n−1
X
v=1
(v+ 1)tv
1 µv
∆λv+ 1
n+ 1 n−1
X
v=1
(v+ 1)tvλv+1∆
1
µv
+tnλn
µn
= Tn,1+Tn,2+Tn,3. (4.2)
Using Minkowski’s inequality,
|Tn|k=|Tn,1+Tn,2+Tn,3|k<3k
|Tn,1|k+|Tn,2|k+|Tn,3|k
. (4.3)
In order to complete the proof of the theorem, it is sufficient to show that
∞
X
n=1
ϕln(k−1)
nl(k−δk)|Tn,r|
k<∞, f or r= 1,2,3. (4.4)
By using H¨older’s inequality and Abel’s transformation, we have
m
X
n=2
ϕln(k−1)
nl(k−δk)|Tn,1|
k =
m
X
n=2
ϕln(k−1)
nl(k−δk)
1
n+ 1 n−1
X
v=1
(v+ 1)tv ∆λv
µv
k
= O(1) m
X
n=2
ϕln(k−1)
nk+l(k−δk)
n−1
X
v=1
v|tv|
|∆λv|
µv
!k
= O(1) m
X
n=2
ϕln(k−1)
nk+l(k−δk)
n−1
X
v=1
v|∆λv| µv
|tv|k n−1
X
v=1
v|∆λv| µv
!k−1
= O(1) m
X
n=2
ϕln(k−1)
n1+l(k−δk)
n−1
X
v=1
v|∆λv| µv
|tv|k
!
= O(1) m
X
v=1
v|∆λv| µv
|tv|k m
X
n=v
ϕln(k−1)
n1+l(k−δk)
!
= O(1) m
X
v=1
v|∆λv| µv
|tv|k
ϕlv(k−1)
− | |k
= O(1) m−1 X v=1 ∆ v|∆λv|
µv v X r=1
ϕlr(k−1)
rl(k−δk)|tr|
k+m|∆λm|
µm v
X
r=1
ϕlr(k−1)
rl(k−δk)|tr|
k
= O(1) m−1
X
v=1
|∆λv|
µv
logv.µv+ m−1
X
v=1
(v+ 1)∆
|∆λv|
µv
logv.µv
+ m|∆λm| µm
logm.µm
= O(1) m−1
X
v=1
|∆λv|logv+ m−1
X
v=1
(v+ 1) 1
µv
|∆2λv|logv.µv
+ m−1
X
v=1
(v+ 1)|∆λv+1|∆
1
µv
logv.µv+m|∆λm|logm
= O(1). (4.5)
m
X
n=2
ϕln(k−1)
nl(k−δk)|Tn,2|
k =
m
X
n=2
ϕln(k−1)
nl(k−δk)
1
n+ 1 n−1
X
v=1
λv+1(v+ 1)tv∆
1 µv k
= O(1) m
X
n=2
ϕln(k−1)
nk+l(k−δk)
n−1
X
v=1
vλv+1|tv|∆
1
µv
!k
= O(1) m
X
n=2
ϕln(k−1)
nk+l(k−δk)
n−1
X
v=1
vλv+1∆
1
µv
|tv|k
n−1
X
v=1
vλv+1∆
1
µv
!k−1
= O(1) m
X
n=2
ϕln(k−1)
n1+l(k−δk)
n−1
X
v=1
vλv+1∆
1
µv
|tv|k
!
= O(1) m−1 X v=1
vλv+1∆
1 µv v X r=1
ϕlr(k−1)
r1+l(k−δk)|tr|
k
+ mλm+1∆
1
µm
m
X
r=1
ϕlr(k−1)
r1+l(k−δk)|tr|
k
= O(1) m−1
X
v=1
λv+1∆
1
µv
µvlogv+ m−1
X
v=1
(v+ 1)∆ λv+1∆
1
µv
! µvlogv
+ mλm+1∆
1
µm
logm.µm
= O(1) m−1
X
v=1
λv+1logv+
m−1
X
v=1
(v+ 1)∆λv+1∆
1
µv
µvlogv
+ m−1
X
v=1
(v+ 1)λv+2∆2
1
µv
µvlogv+mλm+1logm
= O(1)as m→ ∞. (4.6)
m
X
n=1
ϕln(k−1)
nl(k−δk)|Tn,3|
k =
m
X
n=1
ϕln(k−1)
nl(k−δk)
tnλn
µn k
= O(1) m
X
n=1
ϕln(k−1)
nl(k−δk)|tn|
k ∞ X
= O(1) ∞
X
v=1
∆
λ
v
µv
v
X
n=1
ϕln(k−1)
nl(k−δk)|tn|
k
= O(1) ∞
X
v=1 1
µv
∆λvlogv.µv+
∞
X
v=1
λv+1∆
1
µv
logv.µv
= O(1). (4.7)
Collecting (4.2) - (4.7), we have
∞
X
n=1
ϕk−1
n
nk−δk|Tn| k
<∞. (4.8)
Hence proof of the theorem is completed.
5. Corollaries
Corollary 5.1. Let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing sequence satisfying (3.1)-(3.2), (3.5) and following conditions:
m
X
v=1
ϕk−1
v
vk−δk|tv| k
=O(logm.µm)as m→ ∞, (5.1)
m
X
n=v
ϕk−1
n
n1+k−δk =O
ϕk−1
v
vk−δk
. (5.2)
Then the infinite seriesP
anλn/µn isϕ− |C,1;δ|k summable for k≥1and δ≥0.
Proof. By using specific value l= 1 in Theorem 3.1, we will get (5.1) and (5.2). We omit the details as the proof is similar to that of Theorem 3.1 and we use (5.1) and (5.2) instead of (3.3) and (3.4).
Corollary 5.2. Let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing sequence satisfying (3.1)-(3.2), (3.5) and following conditions:
m
X
v=1
ϕk−1
v
vk |tv|
k =O(logm.µ
m)as m→ ∞, (5.3)
m
X
n=v
ϕk−1
n
n1+k =O
ϕk−1
v
vk
. (5.4)
Then the infinite seriesPa
nλn/µn isϕ− |C,1;δ|k summable for k≥1and δ≥0.
Proof. By using specific value l = 1 and δ= 0 in Theorem 3.1, we will get (5.3) and (5.4). We omit the details as the proof is similar to that of Theorem 3.1 and we use (5.3) and (5.4) instead of (3.3)
and (3.4).
Hence theorem 3.1 is a generalization of above Corollaries.
6. Conclusion
The aim of this research article is to formulate the problem of generalization of absolute Ces´aro (ϕ− |C,1;δ;l|k, k ≥1, δ≥0 and l is a real number) summability factor of infinite series which is a motivation for the researchers, interested in theoretical studies of an infinite series. Further, this study has a number of direct applications in rectification of signals in FIR filter (Finite impulse response filter) and IIR filter (Infinite impulse response filter). In a nut shell, the absolute summability methods have vast potential in dealing with the problems based on infinite series.
− | |k
References
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[5] H. S. ¨Ozarslan, On absolute Ces´aro summability factors of infinite series, Com. Math. Anal., 3(1) (2007), 53-56. [6] H. S. ¨Ozarslan and T. Ari, Absolute matrix summability methods, Applied Math. Lett., 24 (2011), 2102-2106. [7] S. K. Saxena, On N¨orlund summability factors of infinite series, Int. J. Math. Analysis, 22(2) (2008), 1097 - 1102. [8] S. M. Mazhar, On (C,1) summability factors of infinite series, Indian J. Math., 14 (1972), 45-48.
[9] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Sci., 7 (1957), 113-141.
[10] M. Balci, Absoluteϕ-summability factors, Comm. Fac. Sci. Univ. Ankara, Ser.A1, 29 (1980), 63-80.
1
Department of Mathematics, National Institute of Technology, Kurukshetra-136119, Haryana, India
2
Faculty of Education, University of Prishtina ”Hasan Prishtina”, Avenue ”Mother Theresa” 5, 10000 Prishtina, Kosovo
∗
