on indexed absolute matrix summability of an infinite series · 2019. 12. 18. · aam: intern. j.,...

274

On Indexed Absolute Matrix Summability of an Infinite Series

*1Lakshmi Narayan Mishra, 2P.K. Das, 3P. Samanta,

4M. Misra and 5U.K. Misra

1School of Advanced Sciences,

Vellore Institute of Technology (VIT)

University, Vellore 632 014

Tamil Nadu, India

[email protected]

3Department of Mathematics

Berhampur University

Odisha, India

[email protected]

2Biswasray Science College

Patapur, Ganjam

Odisha, India

4Department of Mathematics

S.B.R Women’s College

Berhampur, Odisha, India

[email protected]

5National Institute of Science and Technology

Ganjam, Odisha, India

[email protected]

*Corresponding author

Received: June17, 2017; Accepted: February 5, 2018

Abstract

Some results have been established on absolute index Riesz summability factor of an infinite

series. Furthermore, these kind of results can be extended by taking other parameters and an

absolute index matrix summability factor of an infinite series or some weaker conditions. In

the present paper a new result on generalized absolute index matrix summability factor of an

infinite series has been established.

Keywords: Riesz summability; Matrix summability; Index summability method

MSC No.: 40A05, 40D05

Available at

http://pvamu.edu/aam

Appl. Appl. Math.

ISSN: 1932-9466

Vol. 13, Issue 1 (June 2018), pp. 274 - 285

Applications and Applied

Mathematics:

An International Journal

(AAM)

mailto:[email protected]




http://pvamu.edu/aam

AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 275

1. Introduction

Let na be an infinite series and ns be its sequence of partial sums. Let np be a sequence

of non-negative numbers with 0

, as and 0, 1.n

n v i i

v

P p n P p i

Then, the

sequence to sequence transformation

0

1 n

n v v

vn

t p sP

(1)

defines the npN , -mean of the sequence ns generated by the sequence of coefficients np .

The series na is said to be summable ,1,, kpNkn if (Bor, 1985)

1

1

1

.

k

knn n

n n

Pt t

p

(2)

For a lower triangular matrix nkaA , we define the matrices nkaA and nkaA ˆˆ as

follows:

n

nk nv

v k

a a

(3)

and ,...2,1,ˆ,ˆ 000000,1 naaaaaa knnknk .

Clearly ˆandA A are lower semi-matrices. For the sequence of partial sums ns of the series

na let

0

.n

n nk k

k

A s a s

(4)

Then, sAn defines a sequence to sequence transformation ns to nA s . From (4) we have

0

n

n nk k

k

A s a a

.

Subsequently, we get

0

ˆ ,n

n nk k

n

A s a a

where

1n n nA s A s A s .

276 L. N. Mishra et al.

Definition 1.1.

The series na is said to be summable 1,, kpAkn if (Sulaiman, 2003)

. (5)

Definition 1.2.

If n is a sequence of positive real numbers, then, the series na is said to be summable

1,, kpAkn , if (Ozarslan and Karakas, 2015)

k

n

nnn sA

1

1 (6)

and summable 0,1,;, kpAkn , if

k

n

nkk

n sA1

1 . (7)

If we take 0 , then , ;n kA p summability reduces to

knpA, -summability. If we

take

n

nn

p

P , then , n k

A p summability reduces to knpA, -summability. Also, if we take

n

knk

P

pa , then

knpA, summability reduces to knpN , summability. If we take

n

nn

p

P and

n

nnk

P

pa , then

knpA, summability reduces to knpN , -summability.

Definition 1.3.

A sequence of positive numbers nb is said to be almost increasing if there exists a positive

increasing sequence nc and two positive constants A and B such that n n nAc b Bc (Bari and

Steckin, 1956).

2. Known Results

Dealing with summability factors of infinite series, Bor (1996) has proved the following theorem.

1

1

n

k

n

n

n

n sAp

P


Theorem 2.1.

Let np be a sequence of positive numbers such that

0 , asn nP np n . (8)

Let nX be a positive non-decreasing sequence and that there be sequences andn n such

that

n n (9)

0, asn n , (10)

1

n n

n

n X

(11)

and

0(1), asn nX n .

(12)

If

1

, asm

knn m

n n

pt O X m

P

, (13)

where

n

v

vn avn

t11

1, then nna is summable- 1,, kpN

kn .

Extending the above theorem to , , , 1,0 1n kN p k , and taking a weaker condition almost

increasing instead of nX to be a positive non-decreasing sequence, Bor (2010) has established

the following result:

Theorem 2.2.

Let np be a sequence of positive numbers such that

0 , as ,n nP np n (14)

and

1

kk n

n n n

pn O

P p p

. (15)

If nX is an almost increasing sequence satisfying the conditions

0(1), as ,n nX n (16)


2

1

,n n

n

n X

(17)

1

,n

n n

(18)

1

1

, as ,

km

knn m

n n

pt O X m

P

(19)

where

n

v

vn avn

t11

1, then nna is summable- , , , 1,0 1/ .n k

N p k k

Subsequently, Savas and Rhoades (2007) have extended Theorem-B to kA summability factor

establishing the following theorem.

Theorem 2.3.

Let nvaA be a lower triangular matrix with non-negative entries such that

0 1, 0,1,2,...na n (20)

1, , for 1n v nva a n v (21)

1nnn a O (22)

Further, let n be a sequence such that

1

1m

n

n

O

, (23)

1

1m

k

nn n

n

a O

(24)

and ns be a bounded sequence. Then, nna is summable- , 1.k

A k

Very recently, Ozarslan and Karakas (communicated) extended Theorem-C, by establishing the

following theorem.

Theorem 2.4.

Let nvaA be a lower triangular matrix with non-zero diagonal entries such that

0 1, 0,1,2,... ,na n (25)


1, , 1 ,n v nva a for n v (26)

,nnn

n

pa O

P

(27)

nvvvn avOa ˆˆ 1, . (28)

Let nX be an almost increasing sequence and sequence n n

n

p

P

non-increasing. Let n and

n be sequences satisfying (8) to (12). If, in addition the condition

1

1

, as

km

kk nn n m

n n

pt O X m

P

(29)

is satisfied, where nt is as defined in Theorem 2.1., then, the series nna is summable

, , 1n kA p k .

3. Main result

In the present paper we generalize the above theorems to 0,1,;, kpAkn

summability by proving Theorem 3.1.

Theorem 3.1.

Let nvaA be a lower triangular matrix with non-zero diagonal entries such that

0 1, 0,1,2,... ,na n (30)

1, , for 1 ,n v nva a n v (31)

,nnn

n

pa O

P

(32)

nvvvn avOa ˆˆ 1, . (33)


Let nX be an almost increasing sequence and sequences 1k k

n

and

n

n

P

p non-increasing.

Let, also n and n be sequences satisfying (8) to (12). If, in addition, the condition

1

1

, as

km

kk k nn n m

n n

pt O X m

P

(34)

is satisfied, the series nna is summable 0,1,;, kpAkn , where nt is as

defined in Theorem 2.1.

For the proof of the theorem, we require the following lemma.

Lemma 3.1. (Mazhar, 1997)

If nX is an almost increasing sequence satisfying conditions (10) and (11), then we have

1n nn X O (35)

and

1

.n n

n

X

. (36)

Proof of the theorem

Let nT be the nvaA transform of the series nna . Then, using (4), we have

n

v

vvnvn aaT1

ˆ

v

n

v

vnv avv

a

1

ˆ

nnnn

vvnv

n

v

v tnn

atv

v

a11

ˆ1

1

,

(using Abel’s partial summation)

1 1

, 1

1 1

1 1 1ˆ ˆ

n n

nn n nv v v n v v v

v v

n v va t a t a t

n v v

1

, 1 1

1

1ˆ

n

n v v v

v

a t

4,3,2,1, nnnn TTTT , say.


Using (7) and Minkowski’s inequality, in order to complete the proof of the theorem, it is enough

to show that

1

,

1

,kk n

n n i

n

T

for 1,2,3,4.i

We have

1 1

,1

1 1

1m m

k k kk k k k k

n n n nn n n

n n

T O a t

k

nn

k

n

k

n

nm

n

kkn t

P

pO 1

1

11

k

nn

k

n

nm

n

kkn t

P

pO

1

11

1

1 1

1 1 1

(1) (1)

k km n m

k kk k k kv nn v v m n n

n v nv n

p pO t O t

P P

.

Using Abel’s partial summation

mmn

m

n

n XOXO )1()1(1

1

, using (34)

1

1

(1) (1)m

n n m n

n

O X O X

, using (9)

(1) , as ,O m using (36) and (12).

Next,

1 1 1

1 1

,2

2 2 1

ˆ1

km m n

kk k n k

n n n nv v v

n n v

T O a t

1

1 1 11

2 1 1

ˆ ˆ(1)

km n n

k kk k

n nv v v nv

n v v

O a t a

using Holder’s inequality with indices kk & with 1 1

1, 1kk k

1 1

1 1

2 1

ˆ(1)m n

k kk k k

n nn nv v v

n v

O a a t


1

1 11

2 1

ˆ(1)

km n

k kn k nn nv v v

n vn

pO a t

P

, using (32)

1

11

1 1

ˆ(1)

km m

k k n k nv v n nv

v n v n

pO t a

P

1

11

1 1

ˆ(1)

km m

k kk k vv v v nv

v n vv

pO t a

P

vv

k

v

k

v

m

v

k

v

vkkv at

P

pO

1

1

1)1(

k

v

n

v

k

v

vm

v

knv t

P

pO

1

1)1( , using (32)

)1(O , as above.

Also,

k

k

vv

n

v

vn

m

n

knn

k

n

m

n

kkn taOT

1

1

1,

1

2

12,

1

2

1 ˆ1

kn

v

vvvn

m

n

kkn taO

1

1

1,

1

2

1 ˆ)1( , using (9)

11

1

1,

1

2

1

1

1,1 ˆˆ)1(

kk

v

vvn

m

n

n

v

k

vvvnkk

n ataO ,

using Holder’s inequality with 1 1

1, 1kk k

,

1

1 1 11

, ,

2 1 1

ˆ ˆ(1)

km n k

kk k

n n v v v n v v

n v v

O v a t v a

,

using (33)

1 1

1 1

2 1

ˆ(1)m n

kk k k

n nn nv v v

n v

O a v a t

1

1 11

,

2 1

ˆ(1)

km n

kk k nn n v v v

n vn

pO v a t

P


1

11

,

1 1

ˆ(1)

km m

k k k nv v n n v

v n v n

pO v t a

P

1

11

,

1 1

ˆ(1)

km m

k k k nv v n n v

v n vn

pO v t a

P

m

v

vv

k

vv

k

v

vkk

v atvP

pO

1

1

1)1(

k

vv

k

v

vm

v

kk

v tvP

pO

1

1)1(

1

1 1

1

1

1)1(m

v

k

i

k

i

im

i

kk

im

k

i

k

i

iv

i

kk

iv tP

pmt

P

pvO

m

v

mmvv XpmXvO1

)1(

1

1

1

1

)1(m

v

m

v

mmvvvv XmXXvO

)1(O , using (11), (35), and (36).

Finally,

k

vn

v

vvn

m

n

kk

n

k

n

m

n

kk

nv

taOT

1

11,

1

2

1

4,

1

2

1 ˆ)1(

1 1

1

, 1

2 1

ˆ(1)

km n

k k

n n v v v

n v

O a t

, using (334)

1

1 1 11

1

2 1 1

ˆ ˆ(1)

km n n

k kk k

n nv v v nv

n v v

O a t a

,

using Holder’s inequality

1 1

11

1 1

2 1

ˆ(1) 1nn

m nk kk k

n nv v v va

n v

O k a t

1

1 11

1

2 1

ˆ(1)

km n

kk k nn nv v v

n vn

pO a t

P

1

11

1

1 1

ˆ(1)

km m

k k k nv v n nv

v n v n

pO t a

P


1

11

1

1 1

ˆ(1)

km m

kk k vv v v nv

v n vv

pO t a

P

nv

k

vv

k

v

vm

v

kk

v atP

pO 1

1

1

1)1(

k

vv

k

v

vm

v

kk

v tP

pO 1

1

1)1(

)1(O , as above.

This completes the proof of the theorem.

4. Conclusion

If a series is absolute summable, then it is summable by the same summability method. Similarly,

if a series is summable by some indexed summability method, then it is absolutely summable by

the same method. Different authors established many results on indexed summability of different

summability methods of infinite series as well as Fourier series. Some authors established certain

results on indexed summability of factored series. The present paper is one of them. We have

established a result on generalized matrix indexed summability of a factored series by extending

the results of Bor, Savas and Rhoades, as well as Ozarslan and Karakas. For 0 , Theorem 2.4.

is coming as a particular case of our theorem. By taking and 0nn

n

P

p , Theorem 2.3 is

coming as a particular case of our theorem. Furthermore, by taking , ,n vn nv

n n

P pa

p P Theorem

2.2 is coming as a particular case of our theorem and taking nX as a positive non-decreasing

sequence, , and 0n vn nv

n n

P pa

p P , Theorem 2.1. follows from our theorem. The theorem can

be studied for , ; ,n kA p summability of a factor series. This can be further studied by

relaxing the conditions placed on np .

Acknowledgement

The authors are thankful to the reviewers for their valuable suggestions for the improvement of

the paper.

REFERENCES

Bari, N.K. and Steckin, S.B. (1956). Best approximations and differential properties of two

conjugate functions, Trudy Moskov, Mat. Obsc.5, pp. 483-522 (in Russian).


Bor, H. (1985). On two summability methods, Math. Proc. Camb. Philos. Soc., 97, pp. 147-149.

Bor, H. (1996). On knpN , summability factors, Kuwait J. Sci. eng., 23, pp. 1-5.

Bor, H (2010). On an application of almost increasing sequence, Math Common, vol.15, no.1, pp.

57-64.

Bor, H (2010). On an application of almost increasing sequence, Math Common, vol.15, no.1, pp.

57-64.

Mazhar, S.M. (1997). A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica, 25,

pp 233-242.

Ozarslan, H.S and Karakas, A. (2015). A new application of almost increasing sequences,

An.Stint.Univ. At.I. Cuza.Lasi. Mat. (N.S), 61, pp. 153-160.

Ozarslan, H.S and Karakas, A. (2017), A new theorem on , n kA p summability method, under

communication.

Savas, E and Rhoades, B.E (2007). A note on k

A summability factors, Non-linear analysis, 66,

1879-1883.

Sulaiman, W.T (2003). Inclusion theorems for absolute matrix summability methods of an infinite

series, IV, Indian J. Pure. Appl. Math. 34(11), pp 1547-1557.

on indexed absolute matrix summability of an infinite series · 2019. 12. 18. · aam: intern. j.,...

Documents