Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 86757, 6 pages
doi:10.1155/2007/86757
Research Article
Some Geometric Inequalities in a New Banach Sequence Space
M. Mursaleen, Rifat Çolak, and Mikail Et
Received 11 July 2007; Accepted 18 November 2007
Recommended by Peter Yu Hin Pang
The difference sequence space m(φ, p,Δ(r) ), which is a generalization of the space m(φ)
introduced and studied by Sargent (1960), was defined by Çolak and Et (2005). In this
paper we establish some geometric inequalities for this space.
Copyright © 2007 M. Mursaleen et al. 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 and preliminaries
Let Ꮿ denote the space whose elements are finite sets of distinct positive integers. Given
an element σ ∈ Ꮿ, we write c(σ) for the sequence (cn (σ)) such that cn (σ) = 1 for n ∈ σ,
and cn (σ) = 0, otherwise. Further
Ꮿs = σ ∈ Ꮿ :
∞
cn (σ) ≤ s ,
(1.1)
n =1
that is, Ꮿs is the set of those σ whose support has cardinality at most s, where s is a natural
number.
Let w be the set of all real sequences and
Φ = φ = φn ∈ w : φ1 > 0, ∇φk ≥ 0, ∇
φk
≤ 0 (k = 1,2,...) ,
k
(1.2)
where ∇φk = φk − φk−1 . For φ ∈ Φ, Sargent [1] introduced the following sequence space:
m(φ) = x = xn ∈ w : sup sup
s≥1 σ ∈Ꮿs
1
|xn | < ∞ .
φ s n ∈σ
(1.3)
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Journal of Inequalities and Applications
In [2], the space m(φ) has been considered for matrix transformations and in [3] some
of its geometric properties have been considered. Tripathy and Sen [4] extended m(φ) to
m(φ, p),1 ≤ p < ∞. Recently, Çolak and Et [5] defined the space m(φ, p,Δ(r) ) by using the
idea of difference sequences (see [6–8]).
Let r be a positive integer throughout. The operators Δ(r) ,Σ(r) : w→w are defined by
Δ(1) x
Σ(1) x
Δ
Σ
(r)
k
k
(r)
◦Δ
(r)
= (Δx)k = xk − xk+1 ,
= (Σx)k =
∞
xj
(k = 1,2,...),
(1.4)
j =k
(r −1)
=Δ
(1)
◦Δ
,
=Δ
(r)
◦Σ
= id,
(r)
Σ
(r)
=Σ
(1)
(r −1)
◦Σ
,
(r ≥ 2),
the identity on w.
For 0 ≤ p < ∞, the space m(φ, p,Δ(r) ) is defined as follows:
m φ, p,Δ
(r) = x ∈ w : sup
s≥1,σ ∈Ꮿs
1 Δ(r) xn p < ∞ ,
φ s n ∈σ
(1.5)
which is a Banach space (1 ≤ p < ∞) with the norm
r
xm(φ,p,Δ
(r)
1 Δ(r) xn p
|xi | + sup
)=
φ
s≥1,σ ∈Ꮿs s n∈σ
i=1
1/ p
,
(1.6)
and a complete p-normed space (0 < p < 1) with the p-norm
xm p (φ,Δ(r) ) =
r
i =1
1 Δ(r) xn p .
φ
s≥1,σ ∈Ꮿs s n∈σ
|xi | p + sup
(1.7)
In this paper, we will consider the case 1 < p < ∞ to study some geometric properties of
m(φ, p,Δ(r) ). We will examine the Banach-Saks property of type p, strict convexity and
uniform convexity. The space m(φ, p),1 ≤ p < ∞ was defined by Tripathy and Sen [4]
which is in fact m(φ, p,Δ) with Δ replaced by id.
Let 1 < p < ∞. A Banach space X is said to have the Banach-Saks property of type p or
property (BS) p if every weakly null-sequence (xk ) has a subsequence (xki ) such that for
some C > 0, the inequality
n
xki ≤ c(n + 1)1/ p ,
i =0
n = 1,2,3,...,
(1.8)
X
holds.
The property (BS) p for a Cesàro sequence space was considered in [9].
We find uniform convexity and strict convexity of our space through the Gurarii’s
modulus of convexity (see [10, 11]).
For a normed linear space X, the modulus of convexity defined by
βX (ε) = inf 1 − inf αx + (1 − α)y : x, y ∈ S(X), x − y = ε ,
0≤α≤1
(1.9)
M. Mursaleen et al. 3
is called the Gurarii’s modulus of convexity, where S(X) denotes the unit sphere in X and
0 < ε ≤ 2. If 0 < βX (ε) < 1, then X is uniformly convex and if βX (ε) ≤ 1, then X is strictly
convex.
2. Main results
Theorem 2.1. The space m(φ, p,Δ(r) ) has the Banach-Saks property of type p.
Proof. We will prove the case r = 1 and the general case can be followed on the same
lines.
Let (εn ) be a sequence of positive numbers for which ∞
n=1 εn ≤ 1/2. Let (xn ) be a
weakly null sequence in B(m(φ, p,Δ)), the unit ball in m(φ, p,Δ). Set x0 = 0 and z1 =
xn1 = Δx1 . Then there exists s1 ∈ N such that
z1 (i)ei i ∈τ 1
< ε1 ,
(2.1)
m(φ,p,Δ)
w
where τ 1 consists of the elements of σ which exceed s1 . Since xn −→ 0 ⇒ xn →0 coordinatewise, there is n2 ∈ N such that
s1
xn (i)ei i=1
< ε1 ,
when n ≥ n2 .
(2.2)
m(φ,p,Δ)
Set z2 = xn2 = Δx2 . Then there exists s2 > s1 such that
z2 (i)ei i ∈τ 2
< ε2 ,
(2.3)
m(φ,p,Δ)
where τ 2 consists of the elements of σ which exceed s2 . Again using the fact xn →0 coordinatewise, there exists n3 > n2 such that
s2
xn (i)ei i =1
< ε2 ,
when n ≥ n3 .
(2.4)
m(φ,p,Δ)
Continuing this process, we can find two increasing sequences (si ) and (ni ) such that
sj
xn (i)ei i=1
< εj,
when n ≥ n j+1 ,
m(φ,p,Δ)
z j (i)ei i∈τ j
(2.5)
< εj,
m(φ,p,Δ)
where z j = xn j = Δx j and τ j consists of the elements of σ which exceed s j . Note that z j (i)
is a term in the sequence with fixed j and running i.
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Journal of Inequalities and Applications
Since ε j −1 + ε j < 1, we have
1 z j (n) ≤ ε j −1 + ε j < 1,
φ s n ∈σ
(2.6)
for all j ∈ N and s ≥ 1. Hence
n s j −1
sj
=
z
(i)e
+
z
(i)e
+
z
(i)e
j
i
j
i
j
i
n zj
j =1
j =1
m(φ,p,Δ)
i=1
i=s j −1 +1
i=1
j =1 i=s j −1 +1
m(φ,p,Δ)
n +
z
(i)e
j
i
j =1 i∈τ j
sj
p
n z
(i)e
j
i
j =1 i=s j −1 +1
m(φ,p,Δ)
=
n
i=s j −1 +1
+2
m(φ,p,Δ)
sup sup
j =1 s≥1 τ j −1 ∈Ꮿs
n
m(φ,p,Δ)
m(φ,p,Δ)
sj
n ≤
z
(i)e
j
i
j =1
m(φ,p,Δ)
n sj
+
z
(i)e
j
i
n s j −1
≤
z
(i)e
j
i
j =1
i∈τ j
1 z j (i) p
φs i∈τ j −1
n
εj,
j =1
1 z j (i) p ≤ n.
≤
sup sup
φ
s i ∈σ
j =1 s≥1 σ ∈Ꮿs
(2.7)
Therefore by (2.7)
n zj
j =1
≤ n1/ p + 1 ≤ 2n1/ p
(2.8)
m(φ,p,Δ)
since nj=1 ε j ≤ 1/2.
Hence m(φ, p,Δ) has the Banach-Saks property of type p.
Remark 2.2. The above result can also be extended to the case when r =1 and so the
proof should also work for a more general case with Δ replaced by a matrix operator
(transformation).
Theorem 2.3. The Gurarii’s modulus of convexity for the space X = m(φ, p,Δ) is
βX (ε) ≤ 1 − 1 −
where 0 < ε ≤ 2.
p 1/ p
ε
2
,
(2.9)
M. Mursaleen et al. 5
Proof. Let x ∈ m(φ, p,Δ). Then
1 Δxn p
φ
s≥1,σ ∈Ꮿs s n∈σ
xm(φ,p,Δ) = Δxm(φ,p) = x1 + sup
1/ p
.
(2.10)
Let 0 < ε ≤ 2 and consider the sequences
u = (un ) =
v = (vn ) =
1−
1−
p 1/ p ε
ε
,
2
p 1/ p ε
,
2
2
,0,0,... ,
(2.11)
ε
−
,0,0,... .
2
Then Δum(φ,p) = um(φ,p,Δ) = 1, Δvm(φ,p) = vm(φ,p,Δ) = 1, that is, u,v ∈ S(m(φ, p,Δ))
and Δu − Δvm(φ,p) = u − vm(φ,p,Δ) = ε.
For 0 ≤ α ≤ 1,
αu + (1 − α)v p
m(φ,p,Δ)
p
p
p
ε
ε
= αΔu + (1 − α)Δv m(φ,p) = 1 −
+ |2α − 1|
.
2
2
(2.12)
Hence
p
p
ε
inf αu + (1 − α)vm(φ,p,Δ) = 1 −
0≤α≤1
2
.
(2.13)
Therefore, for p ≥ 1
βX (ε) ≤ 1 − 1 −
p 1/ p
ε
2
.
This completes the proof of the theorem.
(2.14)
Corollary 2.4. (i) If ε = 2, then βX (ε) ≤ 1 and hence m(φ, p,Δ) is strictly convex.
(ii) If 0 < ε < 2, then 0 < βX (ε) < 1 and hence m(φ, p,Δ) is uniformly convex.
Remark 2.5. Note that these results are best possible for the time being, that is, they
cannot be readily generalized to the general case because our results also hold for general
matrix transformation.
Acknowledgments
The present paper was completed when Professor Mursaleen visited Firat University
(May-June, 2007). The author is very much grateful to the Firat University for providing hospitalities. This research was supported by FUBAP (The Management Union of the
Scientific Research Projects of Firat University) when the first author visited Firat University under the Project no. 1179.
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Journal of Inequalities and Applications
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M. Mursaleen: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
Email address: mursaleenm@gmail.com
Rifat Çolak: Department of Mathematics, Firat University, 23119 Elazığ, Turkey
Email address: rcolak@firat.edu.tr
Mikail Et: Department of Mathematics, Firat University, 23119 Elazığ, Turkey
Email address: mikailet@yahoo.com