Some further extensions of absolute Cesàro summability for double series

R E S E A R C H

Open Access

Some further extensions of absolute Cesàro

summability for double series

Hamdullah ¸Sevli

*

_{and Ekrem Sava¸s}

*_{Correspondence:}

[email protected]; [email protected] Department of Mathematics, Faculty of Arts and Sciences, ˙Istanbul Commerce University, Üsküdar, ˙Istanbul, Turkey

Abstract

In a recent paper [Sava¸s and Rhoades in Appl. Math. Lett. 22:1462-1466, 2009], the authors extended the result of Flett [Proc. Lond. Math. Soc. 7:113-141, 1957] to double summability. In this paper, we consider some further extensions of absolute Cesàro summability for double series.

MSC: 40F05; 40G05

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

Let∞_m=∞_n=amnbe an infinite double series with real or complex numbers, with partial

sums

smn= m

i= n

j=

aij.

For any double sequence (xmn) we shall define

xmn=xmn–xm+,n–xm,n++xm+,n+.

Denote byA_kthe sequence space defined by

A k=

(smn)∞m,n=: ∞

m= ∞

n=

(mn)k–|amn|k<∞;amn=sm–,n–

fork≥.

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

power conservative fork≥, ifT∈B(A_k),i.e., if

∞

m= ∞

n=

(mn)k–|sm–,n–|k<∞,

then

∞

m= ∞

n=

(mn)k–|tm–,n–|k<∞,

where

tmn= ∞

i= ∞

j=

tmnijsij (m,n= , , . . .).

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

hmnij=

Eα– m–iE

β– n–j

Eα

mE

β

n

,

where

Eα

m=

m+α

m

= (α+m+ ) (m+ )(α+ )≈

mα (α+ ).

The series amnis said to be summable|C(α,β)|k,k≥,α,β> –, if (see [])

∞

m= ∞

n=

(mn)k–σm–,n–αβ k

<∞, ()

whereσmnαβdenotes themn-term of theC(α,β) transform of a sequence (smn);i.e.,

σ_mnαβ=  Eα

mE

β

n m

i= n

j=

Eα– m–iE

β–

n–jsij. ()

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

Theorem  Letα≥γ > –,β≥δ> –,and_mnamnbe a double series with partial

sums smn.If

m

namnis|C(γ,δ)|k-summable,then it is also|C(α,β)|k-summable,k≥.

It then follows that if one setsγ=δ= , thenC(α,β)∈B(A

k) for eachα,β≥. In this

paper, we consider some further extensions of absolute Cesàro summability for double series.

We shall use the following lemmas.

Lemma  Ifθ> –,φ> –,θ–ϕ> andφ–ψ> ,then

∞

m=i ∞

n=j

E_m–iϕ E_n–jψ mnEθ

mE

φ

n

= 

ijEθ_i–ϕ–Eφ_j–ψ–. ()

Proof Forα> –,n≥ since

 Eα

n

=





( –x)α_xn–_dx

and

( –x)–α₌

∞

n=

we obtain

∞

m=i ∞

n=j

E_m–iϕ E_n–jψ mnEθ

mE

φ

n

=

∞

m=i

Eϕ_m–i mEθ

m ∞

n=

Enψ

(n+j)E_n+jφ

=

∞

m=i

Eϕ_m–i mEθ

m ∞

n=

Eψ_n





( –x)φxn+j–dx

=

∞

m=i

Eϕ_m–i mEθ

m 



( –x)φxj– _∞

n=

Eψ_nxn

dx

=





( –x)φ–ψ–_xj–_dx ∞

m=

Eϕm

(m+i)Eθ

m+i

=  jEφ_j–ψ–

∞

m=

Eϕ_m





( –x)θxm+i–dx

=  jEφ_j–ψ–





( –x)θ–ϕ–_xi–_dx

= 

ijEθ_i–ϕ–Eφ_j–ψ–.

For single series, Lemma  due to Chow [].

Lemma  Let≤k≤r<∞andα,β> –.For i,j≥,let

Aij=Aij(α,β) = ∞

m=i ∞

n=j |Eα–

m–i|r/k|E

β– n–j|r/k

mn(Eα

m)r/k(E

β

n)r/k

. ()

Then,if k=r,

Aij=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

O(i–α–_j–β–_{), ifα,β< ,}

O(i–α–_j–_{), ifα< ,β≥,}

O(i–j–β–_{), ifα≥,β< ,}

O(i–_j–_{), ifα,β≥.}

()

If k<r<∞,then

Aij=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

O(i–αkr–j–

βr

k–), if_α<  –k/r,_β<  –k/r, O(i–αkr–_j–kr _{logj), ifα<  –k/r,β=  –k/r,} O(i–αkr–_j–kr_{), ifα<  –k/r,β>  –k/r,} O(i–αkr–_j–

βr

k–_{logi), ifα=  –k/r,β<  –k/r,} O(i–rkj–

r

k logilogj), if_α=  –k/r,_β=  –k/r, O(i–αkr–_j–

r

k_{logi), ifα=  –k/r,β>  –k/r,} O(i–rk_j–

βr

k–_{), ifα>  –k/r,β<  –k/r,} O(i–rk_j–

βr

k–_{logj), ifα>  –k/r,β=  –k/r,} O(i–rkj–

r

k), if_α>  –k/r,_β>  –k/r.

Proof Since|Eα

n| ≤K(α)nαfor allα, andEnα≥M(α)nαforα> –, whereK(α) andM(α)

are positive constants depending only onα, ifk<r<∞, then

Aij=O() ∞

m=i ∞

n=j

m–αkr–n–

βr

k–(m–i+ )

(α–)r

k (n–j+ )

(β–)r k

=O() _i–

m=i

m–αkr–_{(m–i+ )}

(α–)r k ₊

∞

m=i

m–αkr–_{(m–i+ )}

(α–)r k

×

_j–

n=j

n–βkr–_{(n–j+ )}

(β–)r k ₊

∞

n=j

n–βkr–_{(n–j+ )}

(β–)r k

=O()

i–αkr–

i–

m=i

(m–i+ )(α–)k r ₊

∞

m=i

m–kr–

×

j–βkr–

j–

n=j

(n–j+ )(β–)k r +

∞

n=j

n–rk–

=O()i–αkr–_j–

βr k–

i

m= j

n=

m(α–)k r_n

(β–)r

k _+O()i–αkr–_j–r/k

i

m=

m(α–)k r

+O()i–r/kj–βkr–

j

n=

n(β–)k r _+O()(ij)–r/k_.

According as (α–)r_k and (β–)r_k = –, < – or > –, we have (). The case k=ris proved

similarly.

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

Theorem  Let r≥k≥.

(i) It holds thatC(α,β)∈(A

k,Ar)for eachα,β>  –k/r.

(ii) Ifα,β=  –k/rand the condition∞_m=∞_n=(mn)k–_logmlogn|a

mn|k=O()is satisfied thenC(α,β)∈(A

k,Ar).

(iii) If the condition∞_m=∞_n=mk+(–α)r_k–_nk+(–β)_kr–_|a

mn|k=O()is satisfied then

C(α,β)∈(A_k,A_r)for each–k/r<α,β<  –k/r.

(iv) Ifβ=  –k/rand the condition∞_m=∞_n=mk+(–α)rk–_nk–_logn|_amn|k_=O()is

satisfied thenC(α,β)∈(A_k,A

r)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)∈(A

k,Ar)for each–k/r<α<  –k/randβ>  –k/r.

(vi) Ifα=  –k/rand the condition∞_m=∞_n=mk–_nk+(–β)_kr–_logm|a

mn|k=O()is satisfied thenC(α,β)∈(A

k,Ar)for each–k/r<β<  –k/r.

(vii) If the condition∞_m=∞_n=mk–_nk–_logm|a

mn|k=O()is satisfied then

C(α,β)∈(A

k,Ar)for eachα>  –k/r,β<  –k/r.

(viii) If the condition∞_m=∞_n=mk–_nk+(–β)r_k–_|a

mn|k=O()is satisfied then

C(α,β)∈(A_k,A_r)for eachα>  –k/rand–k/r<β<  –k/r.

(ix) Ifβ=  –k/rand the condition∞_m=∞_n=mk–nk–logn|amn|k=O()is satisfied thenC(α,β)∈(A_k,A

Proof We shall show that (σmnαβ)∈Ar,i.e., ∞ m= ∞ n=

(mn)r–σ

αβ

m–,n– r

<∞.

Letτmnαβ denote themn-term of theC(α,β)-transform in terms of (mnamn),i.e.,

τ_mnαβ=  Eα mE β n m i= n j=

Eα_m–i–Eβ_n–j–ijaij.

Forα,β> –, since

τ_mnαβ=mnσ_mnαβ–σ_m,n–αβ –σ_m–,nαβ +σ_{m–,n–}αβ ,

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

∞ m= ∞ n=

(mn)–τ_mnαβr<∞. ()

Using Hölder’s inequality, we have

∞ m= ∞ n=  mnτ

αβ mn r = ∞ m= ∞ n=  mn  Eα mE β n m i= n j=

Eα– m–iE

β– n–jijaij

r ≤ ∞ m= ∞ n=  mn(Eα

m)r(E

β

n)r

_m i= n j=

Eα_m–i–Eβ_n–j–ikjk|aij|k

r/k × _m i= n j=

E_m–iα–Eβ_n–j– (k–)r/k . Since m i= n j=

Eα_m–i–Eβ_n–j–=

Eα_–+

m–

i=

Eα_m–i–

E_β–+

n–

j=

Eβ_n–j–

=

Eα–

 + m– i=

Eα– m–i

E_β–+

n– j=

Eβ_n–j– =

Eα_–+ m i=

Eα_m–i––E_mα––E_α– × E_β–+

n j=

E_n–jβ––Eβ– n –E

β– 

=Eα–

 +Eαm––Eα–E

β–  +E

β

n––E

β–  ,

and using the fact that

Em–α

Eα

m

=O() and E β

n–

Enβ

we obtain

∞

m= ∞

n=



mnτ

αβ

mn r

≤

∞

m= ∞

n=

(Eα

m)(k–)r/k(E

β

n)(k–)r/k

mn(Eα

m)r(E

β

n)r

×

_m

i= n

j=

Eα– m–iE

β– n–ji

k_jk_|a ij|k

r/k

≤

∞

m= ∞

n=

(Eα

m)–r/k(E

β

n)–r/k

mn

×

_m

i= n

j=

Eα– m–iE

β– n–j(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 indicesr/k,r/(r–k), we deduce that

∞

m= ∞

n=

 mnτ

αβ

mn r

≤

∞

m= ∞

n=

(Eα

m)–r/k(E

β

n)–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)∈Ak, we have

∞

m= ∞

n=

 mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–_|a ij|k(ij)r/k

∞

m=i ∞

n=j |Eα–

m–i|r/k|E

β– n–j|r/k

mn(Eα

m)r/k(E

β

n)r/k

.

(i) From Lemma , ifα,β>  –k/r, then

∞

m=i ∞

n=j

(Eα– m–i)r/k(E

β– n–j)r/k

mn(Eα

m)r/k(E

β

n)r/k

= (ij)– 

E r k–

i E

r k–

j

=O(ij)–r/k.

Therefore, for the caseα,β>  –k/r, we have

∞

m= ∞

n=



mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–|aij|k(ij)r/k(ij)–r/k

=O()

∞

i= ∞

j=

(ij)k–|aij|k=O(),

(ii) Ifα,β=  –k/r, from Lemma , then

∞

m=i ∞

n=j |Eα–

m–i|r/k|E

β– n–j|r/k

mn(Eα

m)r/k(E

β

n)r/k

=Oi–kr_j–kr_logilogj.

Hence,

∞

m= ∞

n=



mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–logilogj|aij|k=O().

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

∞

m=i ∞

n=j |Eα–

m–i|r/k|E

β– n–j|r/k

mn(Eα

m)r/k(E

β

n)r/k

=Oi–αkr–_j–

βr k–_,

and then

∞

m= ∞

n=

 mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k+r/k–i–αkr–j–

βr k–|a_ij|k

=O()

∞

i= ∞

j=

ik+(–α)rk–_jk+(–β)rk–|_aij|k_=O().

(iv) Ifβ=  –k/rand –k/r<α<  –k/r, from Lemma , then

∞

m=i ∞

n=j |Eα–

m–i|r/k|E

β– n–j|r/k

mn(Eα

m)r/k(E

β

n)r/k

=Oi–αkr–_j–kr _logj,

therefore, we have

∞

m= ∞

n=



mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–|aij|k(ij)r/ki– αr

k–_j–kr_logj

=O()

∞

i= ∞

j=

ik+(–α)kr–_jk–_logj|_aij|k_=O().

(v) If –k/r<α<  –k/randβ>  –k/r, then

∞

m= ∞

n=



mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–|aij|k(ij)r/ki– αr

k–_j–kr

=O()

∞

i= ∞

j=

ik+(–α)rk–_jk–|_aij|k_=O(),

(vi) Ifα=  –k/rand –k/r<β<  –k/r, then

∞

m= ∞

n=



mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–|aij|k(ij)r/ki–

r k_j–

βr k–_logi

=O()

∞

i= ∞

j=

ik–jk+(–β)kr–_logi|_aij|k_=O(),

by using Lemma .

(vii) Ifα=  –k/randβ>  –k/r, then

∞

m= ∞

n=

 mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–|aij|k(ij)r/ki–

r k_j–kr_logi

=O()

∞

i= ∞

j=

ik–jk–logi|aij|k=O(),

by using Lemma .

(viii) Ifα>  –k/rand –k/r<β<  –k/r, then

∞

m= ∞

n=

 mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–|aij|k(ij)r/ki–

r kj–

βr k–

=O()

∞

i= ∞

j=

ik–jk+(–β)_kr–_|a

ij|k=O(),

by using Lemma .

(ix) Ifα>  –k/randβ=  –k/r, then

∞

m= ∞

n=



mnτ

αβ

mn r

=O()

∞

i= ∞

j=

(ij)k–|aij|k(ij)r/ki–

r k_j–

βr k–_logj

=O()

∞

i= ∞

j=

ik–jk–logj|aij|k=O(),

by using Lemma .

The one-dimensional version of Theorem  appears in []. By (), Theorem  includes the following theorem with the special caser=k.

Theorem  Let k≥.

(i) It holds thatC(α,β)∈B(A_k)for eachα,β≥. (ii) If the condition∞_m=∞_n=mk–α–_nk–β–_|a

mn|k=O()is satisfied then

C(α,β)∈B(A

k)for each– <α< and– <β< .

(iii) If the condition∞_m=∞_n=mk–α–_nk–_|a

mn|k=O()is satisfied then

C(α,β)∈B(A

k)for each– <α< andβ≥.

(iv) If the condition∞_m=∞_n=mk–nk–β–_|a

mn|k=O()is satisfied then

Remark  Theorem  moderates Theorem  of []. Since Holder’s inequality is valid if each of the terms is nonnegative, it should be added the absolute values of the binomial coefficients 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) of Theorem  in the statement of Theorem  of [], for the cases – <α<  and/or – <β< .

Corollary  Letθ_mnα =_Eα

m m

i=Em–iα–sin=C(α, )(smn).

(i) It holds thatC(α, )∈B(A

k)for eachα≥.

(ii) If the condition∞_m=∞_n=mk–α–_nk–_|a

mn|k=O()is satisfied thenC(α, )∈B(Ak) for each– <α< .

Corollary  Letθmnβ =  Eβn

n j=E

β–

n–jsmj= (C, ,β)(smn).

(i) It holds thatC(,β)∈B(A

k)for eachβ≥.

(ii) If the condition∞_m=∞_n=mk–nk–β–_|a

mn|k=O()is satisfied thenC(,β)∈B(Ak) for each– <β< .

Corollary  Letσmn=_{(m+)(n+)}

m i=

n

j=sij= (C, , )(smn).Then C(, )∈B(A_k).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. Both the authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M. Srivastava.

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

References

1. Rhoades, BE: Absolute comparison theorems for double weighted mean and double Cesàro means. Math. Slovaca

48, 285-301 (1998)

2. Sava¸s, E, Rhoades, BE: An inclusion theorem for double Cesàro matrices over the space of absolutelyk-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, Sava¸s, E: On absolute Cesàro summability. J. Inequal. Appl.2009, Article ID 279421 (2009)

7. Sava¸s, 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-2013-144

Cite this article as:¸Sevli and Sava¸s:Some further extensions of absolute Cesàro summability for double series.