R E S E A R C H
Open Access
Logarithmic density and logarithmic
statistical convergence
Mohammed A Alghamdi
1, Momammad Mursaleen
2*and Abdullah Alotaibi
1*Correspondence:
mursaleenm@gmail.com
2Department of Mathematics,
Aligarh Muslim University, Aligarh, 202002, India
Full list of author information is available at the end of the article
Abstract
In this paper we use the idea of logarithmic density to define the concept of logarithmic statistical convergence. We find the relations of logarithmic statistical convergence with statistical convergence, statistical summability (H, 1) introduced by Móricz (Analysis 24:127-145, 2004) and [H, 1]q-summability. We also give subsequence characterization of statistical summability (H, 1).
MSC: 40A05; 40A30
Keywords: asymptotic density; logarithmic density; statistical convergence; statistical summability (H, 1); logarithmic statistical convergence
1 Introduction and preliminaries
The concept of statistical summability (H, ), which is a generalization of statistical con-vergence due to Fast [], has recently been introduced by Móricz []. In this paper we use the idea of logarithmic density to define the concept of logarithmic statistical conver-gence. We find its relation with statistical convergence and statistical summability (H, ). We further define [H, ]q-summability and establish some inclusion relations.
Definition . Let Nbe the set of all natural numbers and let χE denote the
charac-teristic function ofE⊆N. Putdn(E) = n
n
k=χE(k) andδn(E) = ln
n k=
χE(k)
k forn∈N,
where ln=
n
k=/k (n= , , , . . .). The numbersd(E) =lim infn→∞dn(E) and d¯(E) =
lim supn→∞dn(E) are called thelowerandupper asymptotic densityofE, respectively.
Simi-larly, the numbersδ(E) =lim infn→∞δn(E) andδ¯(E) =lim supn→∞δn(E) are called thelower
andupper logarithmic densityofE, respectively. Ifd(E) =d¯(E) =d(E), thend(E) is called theasymptotic densityofE(δ(E) =δ¯(E) =δln(E) is called thelogarithmic densityofE,
re-spectively).
Note that fork= ,ln=
n
k=/k=nand henceδln(E) reduces tod(E).
Now recall the concept of statistical convergence of real sequences (see Fast [] and Fridy []).
Definition . A sequencex= (xk) is said to bestatistically convergenttoLif for every
> ,d({k:|xk–L| ≥}) = . That is,
lim
n
nk≤n:|xk–L| ≥= .
Several extensions, variants and generalizations of this notion have been investigated by various authors, namely [, –].
2 Logarithmic statistical convergence
In this section we define the logarithmic statistical convergence and [H, ]q-summability
and establish some inclusion relations.
Definition . A sequencex= (xk) is said to belogarithmic statistically convergenttoLif
for every> , the set{k:|xk–L| ≥}has logarithmic density zero. That is,
lim
n
ln
k≤n:
k|xk–L| ≥
= . (.)
In this case we write stln-limx=Land we denote the set of all logarithmic statistically
convergent sequences bystln.
Remark . One can say that logarithmic statistical convergence is a special case of weighted statistical convergence [] if pk = k. But this is not exactly true, since for
pk=k, Pn=
n k=pk=
n
k=/k≈logn(n= , , , . . .), and consequently, the definition
of weighted statistical convergence gives thatlimnln|{k≤ln≈logn: k|xk–L| ≥}|= .
So, one can see the difference between this and (.),i.e., in (.) the enclosed set has bigger cardinality.
Definition . Letτn:=l–n
n
k=xk/k, where ln=
n
k=/k≈logn (n= , , , . . .). We
say that x= (xk) is (H, )-summableto Lif the sequence τ = (τn) converges to L, i.e.,
(H, )-limx=L. Ifk= thenln=
n
k=/k=n, and (H, )-summability is reduced to (C, )-summability.
Definition . A sequencex= (xk) is said to be [H, ]q-summable( <q<∞) to the limit
Liflimnln
n
k=k|xk–L|q= , and we write it asxk→L[H, ]q. In this caseLis called the
[H, ]q-limit ofx.
Letq= . Ifk= thenln=
n
k=/k=n, [H, ]q-summability is reduced to strong (C,
)-summability. Also, [H, ]q-summability is a special case of [N¯,pn]q-summability (cf.[])
forpk=k.
Recently, Móricz [] has defined the concept of statistical summability (H, ) as follows.
Definition . A sequencex= (xk) is said to bestatistically summable(H, ) toLif the
sequenceτ= (τn) is statistically convergent toL,i.e.,st-limτ=L=H(st)-limx. We denote
byH(st) the set of all sequences which are statistically summable (H, ) and we call such sequences statistically (H, )-summable sequences.
Remark . Ifx= (xk) is bounded, then st-limk→∞xk=Limplies (C, )-limk→∞xk=L
Theorem . Statistical convergence implies logarithmic statistical convergence but con-verse need not be true.
Proof It is well known that for eachE⊆N,d(E)≤δ(E)≤ ¯δ(E)≤ ¯d(E) (see [], pp.-,
In the following theorem we establish the relation between logarithmic statistical conver-gence and Móricz’s statistical summability (H, ).
Theorem . If a sequence x= (xk)is bounded and logarithmic statistically convergent to
L then it is statistically summable(H, )to L,but not conversely.
Proof Letx= (xk) be bounded and logarithmic statistically convergent toL. WriteK:=
{k∈N:
For converse, we consider the special case whenk= , thenln=nas above. Consider the
sequencex= (xk) defined by
xk=
ifkis odd, ifkis even.
Of course this sequence is not logarithmic statistically convergent. On the other hand,x is (H, )-summable to and hence statistically summable (H, ) to .
This completes the proof of the theorem.
Remark . The above theorem is analogous to Theorem . of [] but this holds for any bounded sequence.
In the next result we establish the inclusion relation between logarithmic statistical con-vergence and [H, ]q-summability.
Theorem . (a)If <q<∞and a sequence x= (xk)is[H, ]q-summable to the limit L,
then it is logarithmic statistically convergent to L.
(b)If(xk)is bounded and logarithmic statistically convergent to L,then xk→L[H, ]q.
Hencex= (xk) is logarithmic statistically convergent toL.
(b) Suppose thatx= (xk) is bounded and logarithmic statistically convergent toL. Then,
for> , we haveδln(K) = . Sincex∈l∞, there existsM> such that|xk–L| ≤M
Remark . The above theorem is analogous to Theorem . of [] but with less restric-tions on the sequencex= (xk).
In the next result we characterize statistical summability (H, ) through the (H, )-summable subsequences.
Theorem . A sequence x= (xk)is statistically summable(H, )to L if and only if there
exists a set K={k<k<· · ·<kn<· · · } ⊆Nsuch thatδ(K) = and(H, )-limxkn=L.
Proof Suppose that there exists a setK={k<k<· · ·<kn<· · · } ⊆Nsuch thatδ(K) =
and (H, )-limxkn=L. Then there is a positive integerNsuch that forn>N,
|τn–L|<. (.)
PutK:={n∈N:|τkn–L| ≥}andK={kN+,kN+, . . .}. Thenδ(K) = andK⊆N–K,
which implies thatδ(K) = . Hencex= (xk) is statistically summable (H, ) toL.
Conversely, letx= (xk) be statistically summable (H, ) toL. Forr= , , , . . . , putKr:=
{j∈N:|τkj–L| ≥/r}andMr:={j∈N:|τkj–L|< /r}. Thenδ(Kr) = and
M⊃M⊃ · · · ⊃Mi⊃Mi+⊃ · · · (.)
and
δ(Mr) = , r= , , , . . . . (.)
Now we have to show that forj∈Mr, (xkj) is (H, )-summable toL. Suppose that (xkj)
is not (H, )-summable toL. Therefore there is> such that|τkj–L| ≥for infinitely
many terms. LetM:={j∈N:|τkj–L|<}and> /r(r= , , , . . .). Then
δ(M) = , (.)
and by (.), Mr⊂M. Henceδ(Mr) = , which contradicts (.) and therefore (xkj) is
(H, )-convergent toL.
This completes the proof of the theorem.
Similarly we can prove the following dual statement.
Theorem . A sequence x= (xk)is logarithmic statistically convergent to L if and only if
there exists a set K={k<k<· · ·<kn<· · · } ⊆Nsuch thatδln(K) = andlimxknn =L.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
Acknowledgements
The authors would like to thank the Deanship of Scientific Research at King Abdulaziz University for its financial support under grant number 151-130-1432.
Received: 26 February 2013 Accepted: 12 July 2013 Published: 26 July 2013 References
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