Some further extensions of absolute Cesàro summability for double series

Sevli and Savas Journal of Inequalities and Applications 2013, 2013:144http://www.journalofinequalitiesandapplications.com/content/2013/1/144

RESEARCH Open Access

Some further extensions of absolute Cesàrosummability for double seriesHamdullah Sevli* and Ekrem Savas

*Correspondence:[email protected];[email protected] of Mathematics,Faculty of Arts and Sciences,Istanbul Commerce University,Üsküdar, Istanbul, Turkey

AbstractIn a recent paper [Savas and Rhoades in Appl. Math. Lett. 22:1462-1466, 2009], theauthors extended the result of Flett [Proc. Lond. Math. Soc. 7:113-141, 1957] to doublesummability. In this paper, we consider some further extensions of absolute Cesàrosummability for double series.MSC: 40F05; 40G05

Keywords: absolute summability; bounded operator; Cesáro matrix; double series;Hausdorff matrix

Let∑∞

m=∑∞

n= amn be an infinite double series with real or complex numbers, with partialsums

smn =m∑i=

n∑j=

aij.

For any double sequence (xmn) we shall define

�xmn = xmn – xm+,n – xm,n+ + xm+,n+.

Denote by Ak the sequence space defined by

Ak =

{(smn)∞m,n= :

∞∑m=

∞∑n=

(mn)k–|amn|k <∞;amn =�sm–,n–

}

for k ≥ .A four-dimensional matrix T = (tmnij : m,n, i, j = , , . . .) is said to be absolutely kth

power conservative for k ≥ , if T ∈ B(Ak), i.e., if

∞∑m=

∞∑n=

(mn)k–|�sm–,n–|k <∞,

then

∞∑m=

∞∑n=

(mn)k–|�tm–,n–|k <∞,

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where

tmn =∞∑i=

∞∑j=

tmnijsij (m,n = , , . . .).

A double infinite Cesáro matrix C(α,β) is a double infinite Hausdorff matrix with entries

hmnij =Eα–m–iE

β–n–j

EαmE

βn

,

where

Eαm =

(m + α

m

)=

�(α +m + )�(m + )�(α + )

≈ mα

�(α + ).

The series∑∑

amn is said to be summable |C(α,β)|k , k ≥ , α,β > –, if (see [])

∞∑m=

∞∑n=

(mn)k–∣∣�σ

αβm–,n–

∣∣k <∞, ()

where σαβmn denotes the mn-term of the C(α,β) transform of a sequence (smn); i.e.,

σαβmn =

EαmE

βn

m∑i=

n∑j=

Eα–m–iE

β–n–j sij. ()

Quite recently, Savaş and Rhoades [] extended the result of Flett [] to double summa-bility. Their theorem is as follows.

Theorem Let α ≥ γ > –, β ≥ δ > –, and∑

m∑

n amn be a double series with partialsums smn. If

∑m

∑n amnis |C(γ ,δ)|k-summable, then it is also |C(α,β)|k-summable, k ≥ .

It then follows that if one sets γ=δ = , then C(α,β) ∈ B(Ak) for each α,β ≥ . In this

paper, we consider some further extensions of absolute Cesàro summability for doubleseries.We shall use the following lemmas.

Lemma If θ > –, φ > –, θ – ϕ > and φ –ψ > , then

∞∑m=i

∞∑n=j

Eϕm–iE

ψ

n–j

mnEθmE

φn=

ijEθ–ϕ–

i Eφ–ψ–j

. ()

Proof For α > –, n≥ since

Eαn=

∫

( – x)αxn– dx

and

( – x)–α =∞∑n=

Eα–n xn,



we obtain

∞∑m=i

∞∑n=j

Eϕm–iE

ψ

n–j

mnEθmE

φn=

∞∑m=i

Eϕm–i

mEθm

∞∑n=

Eψn

(n + j)Eφ

n+j

=∞∑m=i

Eϕm–i

mEθm

∞∑n=

Eψn

∫

( – x)φxn+j– dx

=∞∑m=i

Eϕm–i

mEθm

∫

( – x)φxj–

( ∞∑n=

Eψn xn

)dx

=∫

( – x)φ–ψ–xj–dx

∞∑m=

Eϕm

(m + i)Eθm+i

=

jEφ–ψ–j

∞∑m=

Eϕm

∫

( – x)θxm+i– dx

=

jEφ–ψ–j

∫

( – x)θ–ϕ–xi– dx

=

ijEθ–ϕ–i Eφ–ψ–

j. �

For single series, Lemma due to Chow [].

Lemma Let ≤ k ≤ r <∞ and α,β > –. For i, j ≥ , let

Aij = Aij(α,β) =∞∑m=i

∞∑n=j

|Eα–m–i|r/k|Eβ–

n–j |r/kmn(Eα

m)r/k(Eβn )r/k

. ()

Then, if k = r,

Aij =

⎧⎪⎪⎪⎨⎪⎪⎪⎩O(i–α–j–β–), if α,β < ,O(i–α–j–), if α < ,β ≥ ,O(i–j–β–), if α ≥ ,β < ,O(i–j–), if α,β ≥ .

()

If k < r <∞, then

Aij =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

O(i– αrk –j–

βrk –), if α < – k/r,β < – k/r,

O(i– αrk –j– r

k log j), if α < – k/r,β = – k/r,O(i– αr

k –j– rk ), if α < – k/r,β > – k/r,

O(i– αrk –j–

βrk – log i), if α = – k/r,β < – k/r,

O(i– rk j– r

k log i log j), if α = – k/r,β = – k/r,O(i– αr

k –j– rk log i), if α = – k/r,β > – k/r,

O(i– rk j–

βrk –), if α > – k/r,β < – k/r,

O(i– rk j–

βrk – log j), if α > – k/r,β = – k/r,

O(i– rk j– r

k ), if α > – k/r,β > – k/r.

()



Proof Since |Eαn | ≤ K(α)nα for all α, and Eα

n ≥ M(α)nα for α > –, where K(α) and M(α)are positive constants depending only on α, if k < r <∞, then

Aij =O()∞∑m=i

∞∑n=j

m– αrk –n–

βrk –(m – i + )

(α–)rk (n – j + )

(β–)rk

=O()(i–∑

m=im– αr

k –(m – i + )(α–)r

k +∞∑

m=im– αr

k –(m – i + )(α–)r

k

)

×(j–∑

n=jn–

βrk –(n – j + )

(β–)rk +

∞∑n=j

n–βrk –(n – j + )

(β–)rk

)

=O()(i– αr

k –i–∑m=i

(m – i + )(α–)r

k +∞∑

m=im– r

k –

)

×(j–

βrk –

j–∑n=j

(n – j + )(β–)r

k +∞∑n=j

n– rk –

)

=O()i– αrk –j–

βrk –

i∑m=

j∑n=

m(α–)r

k n(β–)r

k +O()i– αrk –j–r/k

i∑m=

m(α–)r

k

+O()i–r/kj–βrk –

j∑n=

n(β–)r

k +O()(ij)–r/k .

According as (α–)rk and (β–)r

k = –, < – or > –, we have (). The case k = r is provedsimilarly. �

For single series, Lemma due to Mehdi [].We now prove the following theorem.

Theorem Let r ≥ k ≥ .(i) It holds that C(α,β) ∈ (A

k ,Ar ) for each α,β > – k/r.

(ii) If α,β = – k/r and the condition∑∞

m=∑∞

n=(mn)k– logm logn|amn|k =O() issatisfied then C(α,β) ∈ (A

k ,Ar ).

(iii) If the condition∑∞

m=∑∞

n=mk+(–α) rk –nk+(–β) rk –|amn|k =O() is satisfied then

C(α,β) ∈ (Ak ,A

r ) for each –k/r < α,β < – k/r.(iv) If β = – k/r and the condition

∑∞m=

∑∞n=m

k+(–α) rk –nk– logn|amn|k =O() issatisfied then C(α,β) ∈ (A

k ,Ar ) for each –k/r < α < – k/r.

(v) If the condition∑∞

m=∑∞

n=mk+(–α) rk –nk–|amn|k =O() is satisfied then

C(α,β) ∈ (Ak ,A

r ) for each –k/r < α < – k/r and β > – k/r.(vi) If α = – k/r and the condition

∑∞m=

∑∞n=mk–nk+(–β) rk – logm|amn|k =O() is

satisfied then C(α,β) ∈ (Ak ,A

r ) for each –k/r < β < – k/r.(vii) If the condition

∑∞m=

∑∞n=mk–nk– logm|amn|k =O() is satisfied then

C(α,β) ∈ (Ak ,A

r ) for each α > – k/r, β < – k/r.(viii) If the condition

∑∞m=

∑∞n=mk–nk+(–β) rk –|amn|k =O() is satisfied then

C(α,β) ∈ (Ak ,A

r ) for each α > – k/r and –k/r < β < – k/r.(ix) If β = – k/r and the condition

∑∞m=

∑∞n=mk–nk– logn|amn|k =O() is satisfied

then C(α,β) ∈ (Ak ,A

r ) for each α > – k/r.



Proof We shall show that (σαβmn) ∈A

r , i.e.,

∞∑m=

∞∑n=

(mn)r–∣∣�σ

αβm–,n–

∣∣r <∞.

Let ταβmn denote themn-term of the C(α,β)-transform in terms of (mnamn), i.e.,

ταβmn =

EαmE

βn

m∑i=

n∑j=

Eα–m–iE

β–n–j ijaij.

For α,β > –, since

ταβmn =mn

(σαβmn – σ

αβm,n– – σ

αβm–,n + σ

αβm–,n–

),

to prove the theorem, it will be sufficient to show that

∞∑m=

∞∑n=

(mn)–∣∣ταβ

mn∣∣r <∞. ()

Using Hölder’s inequality, we have

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r = ∞∑m=

∞∑n=

mn

∣∣∣∣∣ EαmE

βn

m∑i=

n∑j=

Eα–m–iE

β–n–j ijaij

∣∣∣∣∣r

≤∞∑m=

∞∑n=

mn(Eα

m)r(Eβn )r

{ m∑i=

n∑j=

∣∣Eα–m–i

∣∣∣∣Eβ–n–j

∣∣ikjk|aij|k}r/k

×{ m∑

i=

n∑j=

∣∣Eα–m–i


∣∣}(k–)r/k

.

Since

m∑i=

n∑j=

∣∣Eα–m–i


∣∣ =(∣∣Eα–

∣∣ + m–∑

i=

∣∣Eα–m–i

∣∣)(∣∣Eβ–

∣∣ + n–∑j=

∣∣Eβ–n–j

∣∣)

=(∣∣Eα–

∣∣ +

∣∣∣∣∣m–∑i=

Eα–m–i

∣∣∣∣∣)(∣∣Eβ–

∣∣ +

∣∣∣∣∣n–∑j=

Eβ–n–j

∣∣∣∣∣)

=(∣∣Eα–

∣∣ +

∣∣∣∣∣m∑i=

Eα–m–i–Eα–

m – Eα–

∣∣∣∣∣)

×(∣∣Eβ–

∣∣ +

∣∣∣∣∣n∑j=

Eβ–n–j – Eβ–

n – Eβ–

∣∣∣∣∣)

=(∣∣Eα–

∣∣ + ∣∣Eα

m– – Eα–

∣∣)(∣∣Eβ–

∣∣ + ∣∣Eβn– – Eβ–

∣∣),

and using the fact that

∣∣∣∣Eαm–Eαm

∣∣∣∣ =O() and∣∣∣∣E

βn–

Eβn

∣∣∣∣ =O(),



we obtain

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r

≤∞∑m=

∞∑n=

(Eαm)(k–)r/k(E

βn )(k–)r/k

mn(Eαm)r(E

βn )r

×{ m∑

i=

n∑j=

∣∣Eα–m–i


∣∣ikjk|aij|k}r/k

≤∞∑m=

∞∑n=

(Eαm)–r/k(E

βn )–r/k

mn

×{ m∑

i=

n∑j=

∣∣Eα–m–i


∣∣(ij)–k/r+k/r|aij|k/r(ij)–(r–k)+k(r–k)/r|aij|k(r–k)/r}r/k

.

Applying Hölder’s inequality with indices r/k, r/(r – k), we deduce that

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r ≤∞∑m=

∞∑n=

(Eαm)–r/k(E

βn )–r/k

mn

m∑i=

n∑j=

∣∣Eα–m–i

∣∣r/k∣∣Eβ–n–j

∣∣r/k(ij)k–+r/k|aij|k

×{ m∑

i=

n∑j=

(ij)k–|aij|k}(r–k)/k

.

Since (smn) ∈Ak , we have

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r =O()∞∑i=

∞∑j=

(ij)k–|aij|k(ij)r/k∞∑m=i

∞∑n=j


n–j |r/kmn(Eα

m)r/k(Eβn )r/k

.

(i) From Lemma , if α,β > – k/r, then

∞∑m=i

∞∑n=j

(Eα–m–i)r/k(E

β–n–j )r/k

mn(Eαm)r/k(E

βn )r/k

= (ij)–

Erk –i E

rk –j

=O((ij)–r/k

).

Therefore, for the case α,β > – k/r, we have

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r =O()∞∑i=

∞∑j=

(ij)k–|aij|k(ij)r/k(ij)–r/k

=O()∞∑i=

∞∑j=

(ij)k–|aij|k =O(),

since (smn) ∈Ak .



(ii) If α,β = – k/r, from Lemma , then

∞∑m=i

∞∑n=j


n–j |r/kmn(Eα

m)r/k(Eβn )r/k

=O(i– r

k j– rk log i log j

).

Hence,

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r =O()∞∑i=

∞∑j=

(ij)k– log i log j|aij|k =O().

(iii) If –k/r < α, β < – k/r, from Lemma , then

∞∑m=i

∞∑n=j


n–j |r/kmn(Eα

m)r/k(Eβn )r/k

=O(i– αr

k –j–βrk –),

and then

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r = O()∞∑i=

∞∑j=

(ij)k+r/k–i– αrk –j–

βrk –|aij|k

= O()∞∑i=

∞∑j=

ik+(–α) rk –jk+(–β) rk –|aij|k =O().

(iv) If β = – k/r and –k/r < α < – k/r, from Lemma , then

∞∑m=i

∞∑n=j


n–j |r/kmn(Eα

m)r/k(Eβn )r/k

=O(i– αr

k –j– rk log j

),

therefore, we have

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r =O()∞∑i=

∞∑j=

(ij)k–|aij|k(ij)r/ki– αrk –j– r

k log j

=O()∞∑i=

∞∑j=

ik+(–α) rk –jk– log j|aij|k =O().

(v) If –k/r < α < – k/r and β > – k/r, then

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r = O()∞∑i=

∞∑j=

(ij)k–|aij|k(ij)r/ki– αrk –j– r

k

= O()∞∑i=

∞∑j=

ik+(–α) rk –jk–|aij|k =O(),

by using Lemma .



(vi) If α = – k/r and –k/r < β < – k/r, then

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r = O()∞∑i=

∞∑j=

(ij)k–|aij|k(ij)r/ki– rk j–

βrk – log i

= O()∞∑i=

∞∑j=

ik–jk+(–β) rk – log i|aij|k =O(),

by using Lemma .(vii) If α = – k/r and β > – k/r, then

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r = O()∞∑i=

∞∑j=

(ij)k–|aij|k(ij)r/ki– rk j– r

k log i

= O()∞∑i=

∞∑j=

ik–jk– log i|aij|k =O(),

by using Lemma .(viii) If α > – k/r and –k/r < β < – k/r, then

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r = O()∞∑i=

∞∑j=


βrk –

= O()∞∑i=

∞∑j=

ik–jk+(–β) rk –|aij|k =O(),

by using Lemma .(ix) If α > – k/r and β = – k/r, then

∞∑m=

∞∑n=

mn

∣∣ταβmn

∣∣r = O()∞∑i=

∞∑j=


βrk – log j

= O()∞∑i=

∞∑j=

ik–jk– log j|aij|k =O(),

by using Lemma . �

The one-dimensional version of Theorem appears in []. By (), Theorem includesthe following theorem with the special case r = k.

Theorem Let k ≥ .(i) It holds that C(α,β) ∈ B(A

k) for each α,β ≥ .(ii) If the condition

∑∞m=

∑∞n=mk–α–nk–β–|amn|k =O() is satisfied then

C(α,β) ∈ B(Ak) for each – < α < and – < β < .

(iii) If the condition∑∞

m=∑∞

n=mk–α–nk–|amn|k =O() is satisfied thenC(α,β) ∈ B(A

k) for each – < α < and β ≥ .(iv) If the condition

∑∞m=

∑∞n=mk–nk–β–|amn|k =O() is satisfied then

C(α,β) ∈ B(Ak) for each α ≥ and – < β < .



Remark Theorem moderates Theorem of []. Since Holder’s inequality is valid ifeach of the terms is nonnegative, it should be added the absolute values of the binomialcoefficients in the proof of Theorem of [], when – < α < and/or – < β < . Therefore,if we replace the binomial coefficients with their absolute values, then the inequality ()of [] will be true. So, we should add the conditions, given above in (ii), (iii) and (iv) ofTheorem in the statement of Theorem of [], for the cases – < α < and/or – < β < .

Corollary Let θαmn =

Eαm

∑mi= Eα–

m–isin = C(α, )(smn).(i) It holds that C(α, ) ∈ B(A

k) for each α ≥ .(ii) If the condition

∑∞m=

∑∞n=mk–α–nk–|amn|k =O() is satisfied then C(α, ) ∈ B(A

k)for each – < α < .

Corollary Let θβmn =

Eβn

∑nj= E

β–n–j smj = (C, ,β)(smn).

(i) It holds that C(,β) ∈ B(Ak) for each β ≥ .

(ii) If the condition∑∞

m=∑∞

n=mk–nk–β–|amn|k =O() is satisfied then C(,β) ∈ B(Ak)

for each – < β < .

Corollary Let σmn = (m+)(n+)

∑mi=

∑nj= sij = (C, , )(smn). Then C(, ) ∈ B(A

k).

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe authors contributed equally and significantly in writing this paper. Both the authors read and approved the finalmanuscript.

AcknowledgementsDedicated to Professor Hari M. Srivastava.

Received: 14 December 2012 Accepted: 20 March 2013 Published: 2 April 2013

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48, 285-301 (1998)2. Savas, E, Rhoades, BE: An inclusion theorem for double Cesàro matrices over the space of absolutely k-convergent

double series. Appl. Math. Lett. 22, 1462-1466 (2009)3. Flett, TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math.

Soc. 7, 113-141 (1957)4. Chow, HC: Note on convergence and summability factors. J. Lond. Math. Soc. 20, 459-476 (1954)5. Mehdi, MR: Summability factors for generalized absolute summability. Proc. Lond. Math. Soc. 10, 180-200 (1960)6. Sevli, H, Savas, E: On absolute Cesàro summability. J. Inequal. Appl. 2009, Article ID 279421 (2009)7. Savas, E, Sevli, H, Rhoades, BE: On the Cesàro summability of double series. J. Inequal. Appl. 2008, Article ID 257318

(2008)

doi:10.1186/1029-242X-2013-144Cite this article as: Sevli and Savas: Some further extensions of absolute Cesàro summability for double series.Journal of Inequalities and Applications 2013 2013:144.


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Some further extensions of absolute Cesàro summability for double series