1. ON ABSOLUTE FACTORABLE MATRIX SUMMABILITY METHODS

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ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 8 Issue 1(2016), Pages 1-5.

ON ABSOLUTE FACTORABLE MATRIX SUMMABILITY METHODS

(COMMUNICATED BY HUSEYIN BOR)

MEHMET ALI SARIG ¨OL

Abstract. In this paper we give necessary and sufficient conditions for|C,0|k⇒ |Af|sand|Af|k⇒ |C,0|sfor the case 1< k≤s <∞, where|Af|kis absolute factorable summability. So we obtain some known results.

1. Introduction

LetP_x

v be a given infinite series with partial sums (sn).Byσnαwe denoten.th Ces`aro mean of orderα,α >−1 , of the sequence (sn). The series Pxv is said to be absolutely summable (C, α) with indexk, or simply summable|C, α|_k, k≥1,if (see [6])

∞

X

n=1

nk−1σ α n −σ

α n−1

k

<∞. (1.1)

Sinceσ0

n=sn, the summability|C,0|_k is equivalent to

∞

X

n=1

nk−1|xn| k

<∞. (1.2)

Let (pn) be a sequence of positive real constants withPn=p0+p1+...+pn→ ∞ asn→ ∞. The sequence-to-sequence transformation

tn = 1 Pn

n X

v=0

pvsv

defines the sequence (tn) of the (R, pn) Riesz mean of the sequence (sn), generated by the sequence of coefficients (pn). The series Σxv is then said to be summable |R, pn|k, k≥1, if (see [13])

∞

X

n=1

nk−1|tn−tn−1| k

<∞. (1.3)

2000Mathematics Subject Classification. 40C05,40D25,40F05, 46A45.

Key words and phrases. Absolute Riesz summability; matrix transformation; sequence space. c

2016 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted January 10, 2016. Published January 25, 2016.

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Now, by|Rp|_klet us denote the set of series summable by the summability method |R, pn|k.Then it is easily seen that

|Rp|k=  



a= (an) :

∞

X

n=1

nk−1

pn

PnPn−1 n X

v=1

Pv−1xv k <∞   

, k≥1,

and so it means that the series Σxvis summable|R, pn|kif and only if the sequence

x= (xv)∈ |Rp|_k.

Here, we extend the summability|R, pn|k with factorable matrix as follows: The series Σxv is said to be summable |Af|k, k≥1, if

∞

X

n=1

nk−1

b an n X v=1

avxv k

<∞. (1.4)

A factorable matrixAf = (anv) is one in which each entry

anv =

b

anav, 0≤v≤n

0, v > n (1.5) where (_ban) and (an) are any sequences of real numbers. Note that it is possible to get from it some known notations. For example, if one takes_ban =pn/PnPn−1,

av=Pv−1and_ban = 1/n(n+1), av=v,then|Af|_kare reduced to the summabilities |R, pn|_k and|C,1|_k, respectively.

If A and B are methods of summability, B is said to include A (writtenA⇒B) if every series summable by the method A is also summable by the method B. A and B said to be equivalent (writtenA⇔B) if each methods includes the other.

Problems on inclusion dealing absolute Ces`aro and absolute weighted mean summabilities have been examined by many authors ([2-14]). On this topic, Bor [2] proved sufficient conditions for equivalence of the summabilities|R, pn|kand|C,0|k as follows.

Theorem 1.1. Letk >1 and

∞

X

n=v

nk−1_pk n

Pk nPn−1

=O

_vk−1_pk v−1

Pk v−1

. (1.6)

If

Pn+1≥dPn. (1.7) where d is a constant such thatd >1, then|R, pn|k⇔ |C,0|k.

It has been more recently shown by Sarıg¨ol [10] that the condition (1.6) is omit-ted, and the condition (1.7) is not only sufficient but also necessary for Theorem 1.1 to hold, and also been completed in the following way.

Theorem 1.2. Let 1< k≤s <∞.Then,|R, pn|k⇒ |C,0|sif and only if

m X

v=m−1 1 v

_P vPv−1

pv

k∗!1/k ∗

m+1 X

n=m

ns−1

Ps n−1

!1/s

=O(1), (1.8)

where k* denotes the conjugate index ofk, i.e., 1 k+

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Theorem 1.3. Let 1< k≤s <∞.Then,|C,0|_k ⇒ |R, pn|sif and only if v

X

m=1

P_mk∗₋₁ m

!1/k∗ ∞

X

n=v

_n1−1/s_p n

PnPn−1

s!1/s

=O(1), (1.9)

where k* denotes the conjugate index ofk.

Corollary 1.4. Let k ≥ 1. Then, |C,0|_k ⇔ |R, pn|_k if and only if condition (1.6) is satisfied.

2. _{Main Results}

The aim of this paper is to generalize the above theorems for summability|Af|_k. Now we prove the following theorems.

Theorem 2.1. Let 1< k≤s <∞and A be a factorable matrix given by (1.5) such that_ban.an6= 0 for all n. Then, |Af|_k|⇒ |C,0|_sif and only if

m X

v=m−1 1

v|_bav| k∗

!1/k∗ m+1 X

n=m

ns−1

|an|s !1/s

=O(1), (2.1)

where k* denotes the conjugate index of k, i.e., 1 k +

1 k∗ = 1

Theorem 2.2. Let 1< k≤s <∞and A be a factorable matrix given by (1.5). Then,|C,0|_k⇒ |Af|_s if and only if

m X

v=1 1 v|av|

k∗

!1/k∗ _∞ X

n=m

ns−1|_ban|s !1/s

=O(1), (2.2)

where k* denotes the conjugate index ofk.

Now Theorem 2.1 and Theorem 2.2 immediately give the following result. Corollary 2.3. Let 1 < k < ∞ and A be a factorable matrix given by (1.5) such that_ban.an 6= 0 for all n. Then,|C,0|k ⇔ |Af|_s if and only if the conditions (2.1) and (2.2) with k=sare satisfied.

Before proving theorems we recall a result of Bennett [1] thatT :`k→`sif and only if

m X

v=1

ckv∗

!1/k∗ ∞

X

n=m

bsn !1/s

=O(1), (2.3)

whereT = (tnv) =bncv is a factorable matrix with nonegative entricebncv. Proof of Theorem 2.1. Letx∗

n=n1/s ∗

xn and A∗n(x) =n1/k ∗

An(x),where

An(x) =_ban n X

v=1

avxv, n≥1. (2.4)

Then, Σxn is summable |Af|_k and|C,0|s iffA∗(x)∈ lk andx∗ ∈ls, respectively. On the other hand, it can be written from (2.4) that

xn = 1 an

_A n(x) b

an

−An−1(x) b

an−1

(2.5)

and so

x∗_n= n 1/s∗

an

n−1/k∗A∗n(x)

b

an

−(n−1)

−1/k∗

A∗_n₋₁(x)

b

an−1

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which gives us

x∗_n=

∞

X

v=1

tnvA∗v(x),

where

tnv =    

  

n1/s∗ an

−(n−1)−1/k ∗

b

an−1

, v=n−1 n1/s∗

an

n−1/k∗

b

an

, v=n. 0, v6=n−1, n

(2.6)

Then,|Af|_k |⇒ |C,0|_sif and only if

∞

X

n=1 |A∗n(x)|

k

<∞=⇒

∞

X

n=1 |x∗n|

s

<∞, i.e., T :`k →`s,

whereT is the matrix whose entries are defined by (2.6). Therefore, applying (2.3) to the matrixT, we have that|Af|_k |⇒ |C,0|s iff the condition (2.1) holds, which completes the proof.

Proof of Theorem 2.2. Letn≥1 andx∗_n=n1/k∗_x

nandA∗n(x) =n1/s ∗

An(x), whereAn(x) is given by (2.4). Then,

A∗_n(x) =n1/s∗_ban n X

v=1

v−1/k∗avx∗v= n X

v=1

hnvx∗v

where

hnv =

n1/s∗ b

anv−1/k ∗

av, 1≤v≤n 0, v > n.

Since the reminder of the proof is similar to the above, so it can be omitted.

References

[1] G. Bennett, Some elemantery inequalities, Quart. J. Math. Oxford 38 (1987), 401-425. [2] H. Bor, H., A new result on the high indices theorem, Analysis 29 (2009), 403-405.

[3] H. Bor and B. Kuttner, On the necessary conditions for absolute weighted arithmetic mean summability factors, Acta. Math. Hungar. 54 (1989), 57-61.

[4] L. S. Bosanquet, Review of [5],Math. Reviews, MR0034861 (11,654b) (1950).

[5] G. Sunouchi, Notes on Fourier Analysis, 18, absolute summability of a series with constant terms, Tohoku Math. J., 1(1949). 57-65.

[6] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957), 113-141.

[7] L. McFadden, Absolute N¨orlund summability, Duke Math. J., 9 (1942), 168-207.

[8] S. M. Mazhar, On the absolute summability factors of infinite series, Tohoku Math. J.,23 (1971), 433-451.

[9] M. R. Mehdi, Summability factors for generalized absolute summability I, Proc. London Math. Soc.(3), 10 (1960), 180-199.

[10] M. A. Sarıg¨ol, Characterization of summability methods with high indices, Math. Slovaca 63 (2013), No. 5, 1-6.

[11] M. A. Sarıg¨ol, On inclusion relations for absolute weighted mean summability, J. Math. Anal. Appl. , 181 (3), (1994), 762-767.

[12] C. Orhan and M. A. Sarıg¨ol, On absolute weighted mean summability, Rocky Moun. J. Math. 23 (3), (1993), 1091-1097.

[13] M. A. Sarıg¨ol, On absolute weighted mean summability methods, Proc. Amer. Math. Soc., 115 (1), (1992), 157-160.

[14] M. A. Sarıg¨ol, Necessary and sufficient conditions for the equivalence of the summability methodsN , pn

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Mehmet Ali Sarıg¨ol, Department of Mathematics, Pamukkale University, Denizli 20070, Turkey