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Logarithmic density and logarithmic statistical convergence
Advances in Difference Equations volume 2013, Article number: 227 (2013)
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 introduced by Móricz (Analysis 24:127-145, 2004) and -summability. We also give subsequence characterization of statistical summability .
1 Introduction and preliminaries
The concept of statistical summability , which is a generalization of statistical convergence 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 convergence. We find its relation with statistical convergence and statistical summability . We further define -summability and establish some inclusion relations.
Definition 1.1 Let ℕ be the set of all natural numbers and let denote the characteristic function of . Put and for , where (). The numbers and are called the lower and upper asymptotic density of E, respectively. Similarly, the numbers and are called the lower and upper logarithmic density of E, respectively. If , then is called the asymptotic density of E ( is called the logarithmic density of E, respectively).
Note that for , and hence reduces to .
Definition 1.2 A sequence is said to be statistically convergent to L if for every , . That is,
2 Logarithmic statistical convergence
In this section we define the logarithmic statistical convergence and -summability and establish some inclusion relations.
Definition 2.1 A sequence is said to be logarithmic statistically convergent to L if for every , the set has logarithmic density zero. That is,
In this case we write and we denote the set of all logarithmic statistically convergent sequences by .
Remark 2.1 One can say that logarithmic statistical convergence is a special case of weighted statistical convergence  if . But this is not exactly true, since for , (), and consequently, the definition of weighted statistical convergence gives that . So, one can see the difference between this and (2.1), i.e., in (2.1) the enclosed set has bigger cardinality.
Definition 2.2 Let , where (). We say that is -summable to L if the sequence converges to L, i.e., .
If then , and -summability is reduced to -summability.
Definition 2.3 A sequence is said to be -summable () to the limit L if , and we write it as . In this case L is called the -limit of x.
Let . If then , -summability is reduced to strong -summability. Also, -summability is a special case of -summability (cf. ) for .
Recently, Móricz  has defined the concept of statistical summability as follows.
Definition 2.4 A sequence is said to be statistically summable to L if the sequence is statistically convergent to L, i.e., . We denote by the set of all sequences which are statistically summable and we call such sequences statistically -summable sequences.
Remark 2.2 If is bounded, then implies (see ). The converse is obviously not true, e.g., is -summable to but not statistically convergent. However, for bounded sequences, statistical convergence to some number is equivalent to strong Cesàro summability to the same number. But for logarithmic statistical convergence the situation is different (see ).
Theorem 2.1 Statistical convergence implies logarithmic statistical convergence but converse need not be true.
Proof It is well known that for each , (see , pp.70-75, pp.95-96). Hence if exists, then also exists and . Hence statistical convergence implies logarithmic statistical convergence.
Take () and . If for , then
Since (, ), we get
Hence and consequently , i.e., does not exist. Define the sequence by
Since , we have . But does not exist because () and hence does not exist.
This completes the proof. □
3 Main results
In the following theorem we establish the relation between logarithmic statistical convergence and Móricz’s statistical summability .
Theorem 3.1 If a sequence is bounded and logarithmic statistically convergent to L then it is statistically summable to L, but not conversely.
Proof Let be bounded and logarithmic statistically convergent to L. Write . Then
as , which implies that as . That is, x is -summable to L and hence statistically summable to L.
For converse, we consider the special case when , then as above. Consider the sequence defined by
Of course this sequence is not logarithmic statistically convergent. On the other hand, x is -summable to 1 and hence statistically summable to 1.
This completes the proof of the theorem. □
Remark 3.1 The above theorem is analogous to Theorem 2.1 of  but this holds for any bounded sequence.
In the next result we establish the inclusion relation between logarithmic statistical convergence and -summability.
Theorem 3.2 (a) If and a sequence is -summable to the limit L, then it is logarithmic statistically convergent to L.
If is bounded and logarithmic statistically convergent to L, then .
Proof (a) If and , then
as . That is, and so , where . Hence is logarithmic statistically convergent to L.
Suppose that is bounded and logarithmic statistically convergent to L. Then, for , we have . Since , there exists such that (). We have
Now if then . For , we have
as , since . Hence .
This completes the proof of the theorem. □
Remark 3.2 The above theorem is analogous to Theorem 2.2 of  but with less restrictions on the sequence .
In the next result we characterize statistical summability through the -summable subsequences.
Theorem 3.3 A sequence is statistically summable to L if and only if there exists a set such that and .
Proof Suppose that there exists a set such that and . Then there is a positive integer N such that for ,
Put and . Then and , which implies that . Hence is statistically summable to L.
Conversely, let be statistically summable to L. For , put and . Then and
Now we have to show that for , is -summable to L. Suppose that is not -summable to L. Therefore there is such that for infinitely many terms. Let and (). Then
and by (3.2), . Hence , which contradicts (3.3) and therefore is -convergent to L.
This completes the proof of the theorem. □
Similarly we can prove the following dual statement.
Theorem 3.4 A sequence is logarithmic statistically convergent to L if and only if there exists a set such that and .
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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.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.