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# A new note on generalized absolute matrix summability

HS Özarslan* and T Ari

Author Affiliations

Department of Mathematics, Erciyes University, Kayseri, 38039, Turkey

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Journal of Inequalities and Applications 2012, 2012:166 doi:10.1186/1029-242X-2012-166

 Received: 3 June 2012 Accepted: 13 July 2012 Published: 27 July 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper gives necessary and sufficient conditions in order that a series should be summable , , whenever is summable . Some new results have also been obtained.

MSC: 40D25, 40F05, 40G99.

##### Keywords:
summability factors; absolute matrix summability; infinite series

### 1 Introduction

Let be a given infinite series with the partial sums . Let be a sequence of positive numbers such that

(1)

The sequence-to-sequence transformation

(2)

defines the sequence of the Riesz means of the sequence generated by the sequence of coefficients (see [3]). The series is said to be summable , if (see [1])

(3)

Let be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence to , where

(4)

The series is said to be summable , if (see [7])

(5)

where

If we take , then summability is the same as summability.

Before stating the main theorem we must first introduce some further notations.

Given a normal matrix , we associate two lover semimatrices and as follows:

(6)

and

(7)

It may be noted that and are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we have

(8)

and

(9)

If A is a normal matrix, then will denote the inverse of A. Clearly if A is normal, then is normal and has two-sided inverse , which is also normal (see [2]).

Sarıgöl [6] has proved the following theorem for summability method.

Theorem ASuppose thatandare positive sequences withandas. Thenis summable, wheneveris summable, if and only if

(10)

(11)

(12)
provided that

Theorem BThesummability implies the, , summability if and only if the following conditions hold:

(13)

(14)

(15)

where

and we regarded that the above series converges for eachvand Δ is the forward difference operator.

It may be remarked that the above theorem has been proved by Orhan and Sarıgöl [5].

Lemma ([4])

if and only if

(16)

for the cases, wheredenotes the set of all matricesAwhich mapinto.

### 2 Main theorem

The aim of this paper is to generalize Theorem A for the and summabilities. Therefore we shall prove the following theorem.

TheoremLet, andbe two positive normal matrices such that

(17)

(18)

Then, in order thatis summablewheneveris summable, it is necessary that

(19)

(20)

(21)

Also (19)-(21) and

(22)

(23)

(24)

are sufficient for the consequent to hold.

It should be noted that if we take and , then we get Theorem A. Also if we take , then we get Theorem B.

Proof of the Theorem Necessity. Let and denote A-transform and B-transform of the series and , respectively. Then, by (8) and (9), we have

(25)

For , we define

Then it is routine to verify that these are BK-spaces, if normed by

(26)

and

(27)

respectively. Since is summable implies is summable , by the hypothesis of the theorem,

Now consider the inclusion map defined by . This is continuous, which is immediate as A and B are BK-spaces. Thus there exists a constant M such that

(28)

By applying (25) to ( is the vth coordinate vector), we have

and

So (26) and (27) give us

and

Hence it follows from (28) that

Using (17), we can find

The above inequality will be true if and only if each term on the left-hand side is . Taking the first term,

then

which verifies that (19) is necessary. Using the second term, we have

which is condition (20). Now, if we apply (22) to , we have

and

respectively. Hence

Hence it follows from (28) that

Using (18) we can find

which is condition (21).

Sufficiency. We use the notations of necessity. Then

(29)

which implies

(30)

In this case

On the other hand, since

by (22), we have

(31)

By considering the equality

where is the Kronecker delta, we have that

and so

Let

Since

to complete the proof of Theorem, it is sufficient to show that

Then

where

Now

is equivalently

(32)

by Lemma. But it follows from conditions (20), (21) and (23) that

Finally,

where

Now

is equivalently

(33)

by Lemma. But it follows from conditions (21) and (24) that

Therefore, we have

This completes the proof of the Theorem. □

### References

1. Bor, H: On the relative strength of two absolute summability methods. Proc. Am. Math. Soc.. 113, 1009–1012 (1991)

2. Cooke, RG: Infinite Matrices and Sequence Spaces, Macmillan & Co., London (1950)

3. Hardy, GH: Divergent Series, Oxford University Press, Oxford (1949)

4. Maddox, IJ: Elements of Functional Analysis, Cambridge University Press, Cambridge (1970)

5. Orhan, C, Sarıgöl, MA: On absolute weighted mean summability. Rocky Mt. J. Math.. 23(3), 1091–1098 (1993). Publisher Full Text

6. Sarıgöl, MA: On the absolute Riesz summability factors of infinite series. Indian J. Pure Appl. Math.. 23(12), 881–886 (1992)

7. Tanovic̆-Miller, N: On strong summability. Glas. Mat.. 34, 87–97 (1979)