Volume 2009, Article ID 279421,7pages doi:10.1155/2009/279421
Research Article
On Absolute Ces `aro Summability
Hamdullah S¸evli
1and Ekrem Savas¸
21Department of Mathematics, Faculty of Arts and Sciences, Y ¨uz ¨unc ¨u Yıl University, 65080 Van, Turkey 2Department of Mathematics, ˙Istanbul Ticaret University, ¨Usk ¨udar 36472, ˙Istanbul, Turkey
Correspondence should be addressed to Hamdullah S¸evli,[email protected]
Received 14 July 2008; Accepted 7 June 2009
Recommended by L´aszl ´o Losonczi
Denote byAkthe sequence space defined byAk{sn:∞n1nk−1|an|k <∞, an sn−sn−1}for
k≥1. In a recent paper by E. Savas¸ and H. S¸evli2007, they proved every Ces`aro matrix of order
α, forα >−1,C, α∈BAkfork≥1. In this paper, we consider a further extension of absolute
Ces`aro summability.
Copyrightq2009 H. S¸evli and E. Savas¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Letav denote a series with partial sumssn. For an infinite matrixT,tn, thenth term of theT-transform ofsnis denoted by
tn
∞
v0
tnvsv. 1.1
A seriesavis said to be absolutelyT-summable ifn|Δtn−1|<∞, whereΔis the forward
difference operator defined byΔtn−1 tn−1−tn. Papers dealing with absolute summability date back at least as far as Fekete1.
A sequencesnis said to be of bounded variationbvif
n|Δsn|<∞. Thus, to say that a series is absolutely summable by a matrixTis equivalent to saying that theT-transform the sequence is in bv. Necessary and sufficient conditions for a matrixT : bv → bvare known.See, e.g., Stieglitz and Tietz2.
Letσα
said to be absolutelyC, αsummable of orderk≥1, writtenanis summable|C, α|k, if
∞
n1
nk−1σnα−1−σnα k
<∞. 1.2
He then proved the following inclusion theorem.
Theorem 1.1 see3. If a seriesan is summable|C, α|k, then it is summable|C, β|r for each r≥k≥1,α >−1,β > α1/k−1/r.
It then follows that if one choosesr k, then a seriesan, which is|C, α|ksummable, is also|C, β|ksummable fork≥1,β > α >−1.
Absolute Abel summability, written as|A|, was defined by Whittaker4as follows. A seriesanis said to be summable|A|if the seriesanxnis convergent for 0≤x <1 and its sum-functionφxsatisfies the condition:
1
0
φxdx <∞. 1.3
In the same paper, Flett extended this result to indexk by replacing condition1.3by the condition:
1
0
1−xk−1φxkdx <∞. 1.4
Thus the seriesan is said to be summable|A|k,k ≥ 1,if the series
anxn is convergent for 0 ≤ x < 1 and its sum-function φx satisfies condition 1.4. He then showed that summability|A|kis a weaker property than summability|C, α|kfor anyα >−1.
2. The Space
A
kLetanbe a series with partial sumssn. Denote byAkthe sequence space defined by
Ak
sn:
∞
n1
nk−1|an|k<∞, ansn−sn−1
. 2.1
If one setsα0 in the inclusion statement involvingC, αandC, β, then one obtains the fact thatC, β∈BAkfor eachβ >0, whereBAkdenotes the algebra of all matrices that mapAktoAk.
In 1970, using the same definition as Flett, Das 5 defined such a matrix to be absolutely kth power conservative for k ≥ 1, if T ∈ BAk; that is, if sn is a sequence satisfying
∞
n1
nk−1|sn−sn−1|k<∞, 2.2
then
∞
n1
nk−1|tn−tn−1|k<∞. 2.3
Fork 1, condition2.2guarantees the convergence ofsn. Note that whenk > 1,2.2 does not necessarily imply the convergence ofsn. For example, take
sn n
v1
1
vlogv1. 2.4
Then2.2holds butsndoes not converge. Thus, since the limit ofsnneeds not to exist, we cannot introduce the concept of absolutekth power regularity whenk >1.
In that same paper, Das proved that every conservative HausdorffmatrixH∈BAk, which contains as a special case the fact thatC, β∈BAkforβ >0. We know that ifβ≥0, thenC, βis regular, and ifβ <0, thenC, βis neither conservative nor regular. In6, the result of Flett and Das was extended by the following theorem.
Theorem 2.1see6. It holds thatC, α∈BAkfor eachα >−1.
Remark 2.2. In6, when−1 < α < 0 it should be added the condition
∞
n1
nk−α−1|an|kO1. 2.5
in the statement ofTheorem 2.1. Also, it should be added the absolute values of the binomial coefficients in the proof ofTheorem 2.1for the case−1 < α < 0.
Since summability|A|kis a weaker property than summability|C, α|kfor anyα >−1,
fromTheorem 2.1, we obtain the following theorem.
Theorem 2.3. Ifsn∈ Akthen
anis summable|A|k,k≥1.
3. The Main Results
In this paper we consider a further extension of absolute Ces`aro summability. If one sets
α 0 inTheorem 1.1, then one obtains the fact thatC, β ∈ Ak,Arfor eachr ≥ k ≥ 1,
We will use the following Lemma.
Lemma 3.1see7. Ifθ >−1andθ−ϕ >0,then
∞
nv Eϕn−v
nEθ n
1
vEvθ−ϕ−1 , Eθ
n Γ
θn1
Γn1Γθ1≈
nθ
Γθ1. 3.1
We now prove the following theorem.
Theorem 3.2. Letr≥k≥1.
iIt holds thatC, α∈Ak,Arfor eachα >1−k/r. iiIfα 1−k/r and the conditionn∞1nk−1logn|an|k
O1is satisfied thenC, α ∈ Ak,Ar.
iiiIf the condition∞n1nkr/k1−α−2|an|k O1is satisfied thenC, α ∈ Ak,Arfor each−k/r < α <1−k/r.
Proof. Letσα
n denote thenth term of the Ces´aro mean of orderαof a sequencesn; that is,
σnα 1 Eα
n n
v0
Enα−−1vsv. 3.2
We will show thatσα
n∈ Ar; that is,
∞
n1
nr−1σnα−σnα−1 r
<∞. 3.3
Letτα
n denote thenth term of the Ces´aro mean of orderαα >−1of the sequencenan; that is,
τnα 1 Eα
n n
v1
Eαn−−1vvav. 3.4
Sinceτα
n nσnα−σnα−1 see8, condition3.3can also be written as
∞
n1
1 n|τ
α
n|r <∞. 3.5
It follows from H ¨older’s inequality that
∞
n1
1 n|τ
α n|r
∞
n1
1 n 1 Eα n n
v1
Eαn−−1vvav r
≤∞ n1
1 nEnαr
n
v1
Eα−1
n−vvk|av|k r/k
× n
v1
Eα−1
n−v
k−1r/k .
Since
n
v1
Eα−1
n−vEα0−1 n−1
v1
Eα−1
n−vEα0−1
n−1
v1
Eαn−−1v
Eα−1 0 n
v0
Enα−−1v−Enα−1−Eα0−1
Eα0−1E αn−1−Eα0−1,
3.7
and using the fact that
Eαn−1
Eαn
O1, 3.8
we obtain
∞
n1
1 n|τ
α n|r ≤
∞
n1
Eα nk−1r/k nEnαr
n
v1
Eα−1
n−vvk|av|k r/k
≤∞ n1
Eα n−r/k
n n
v1
Eα−1
n−vv1−k/rk 2/r
|av|k2/rv−r−kkr−k/r|av|kr−k/r r/k
.
3.9
Applying H ¨older’s inequality with indicesr/k,r/r−k, we deduce that
∞
n1
1 n|τ
α n|r ≤
∞
n1
Eα n−r/k
n n
v1
Eα−1
n−v r/k
vk−1r/k|av|k n
v1
vk−1|av|k r−k/k
. 3.10
Sincesn∈ Ak,we have
∞
n1
1 n|τ
α
n|r O1
∞
v1
vk−1|av|kvr/k∞ nv
Eα−1 n−v
r/k
nEα nr/k
. 3.11
FromLemma 3.1, ifα >1−k/r,then
∞
nv
Eα−1 n−v
r/k
nEα nr/k
Ov−r/k, 3.12
therefore
∞
n1
1 n|τ
α
n|r O1
∞
v1
Ifα1−k/r,thenSee Lemma 5 of9.
∞
nv Eα−1
n−v r/k
nEα nr/k
Ov−r/klogv, 3.14
and then
∞
n1
1 n|τ
α
n|r O1
∞
v1
vk−1logv|av|kO1. 3.15
If−k/r < α <1−k/r,thenSee Lemma 5 of9
∞
nv Eα−1
n−v r/k
nEαnr/k
Ov−αr/k−1, 3.16
hence
∞
n1
1 n|τ
α
n|r O1
∞
v1
vkr/k1−α−2|av|kO1. 3.17
Theorem 3.2includesTheorem 2.1with the special caserk.
Theorem 3.3. Ifsn∈ Ak, then
anis summable|A|r, r≥k≥1.
Proof. Using the fact that the summability|A|kis a weaker property than summability|C, α|k for anyα >−1, then the proof follows fromTheorem 3.2.
Now we give some negative results.
Corollary 3.4. Letk < r.Thensn∈ Ar does not imply that the series
anis summable|A|k.
Proof. Let p be any number such that k < p < r and let an 1/nlogn1/p. Then, we have sn ∈ Ar. As in the proof of Flett, since
1 01−x
k−1|
φx|kdxis divergent, an is not summable|A|k.
Corollary 3.5. Letk < r.ThenC, α/∈Ar,Akfor anyα >−1.
Proof. The proof followsTheorem 3.3andCorollary 3.4.
References
1 M. Fekete, “Zur theorie der divergenten reihen,”Math. `es Termezs `Ertesit¨o (Budapest), vol. 29, pp. 719– 726, 1911.
2 M. Stieglitz and H. Tietz, “Matrixtransformationen von Folgenr¨aumen. Eine Ergebnis ¨ubersicht,”
Mathematische Zeitschrift, vol. 154, no. 1, pp. 1–16, 1977.
3 T. M. Flett, “On an extension of absolute summability and some theorems of Littlewood and Paley,”
Proceedings of the London Mathematical Society, vol. 7, pp. 113–141, 1957.
4 J. M. Whittaker, “The absolute summability of a series,” Proceedings of the Edinburgh Mathematical Society, vol. 2, pp. 1–5, 1930.
5 G. Das, “A Tauberian theorem for absolute summability,”Proceedings of the Cambridge Philosophical Society, vol. 67, pp. 321–326, 1970.
6 E. Savas¸ and H. S¸evli, “On extension of a result of Flett for Ces`aro matrices,”Applied Mathematics Letters, vol. 20, no. 4, pp. 476–478, 2007.
7 H. C. Chow, “Note on convergence and summability factors,”Journal of the London Mathematical Society, vol. 29, pp. 459–476, 1954.
8 E. Kogbetliantz, “Sur les s´eries absolument sommables par la m´ethode des moyennes arithm´etiques,”
Bulletin des Sciences Math´ematiques, vol. 49, pp. 234–256, 1925.