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On a new application of almost increasing sequences
Journal of Inequalities and Applications volume 2013, Article number: 13 (2013)
In (Bor in Int. J. Math. Math. Sci. 17:479-482, 1994), Bor has proved the main theorem dealing with summability factors of an infinite series. In the present paper, we have generalized this theorem on the summability factors under weaker conditions by using an almost increasing sequence instead of a positive non-decreasing sequence.
MSC:40D15, 40F05, 40G99.
Let be a given infinite series with the partial sums . We denote by the n th mean of the sequence . The series is said to be summable , , if (see )
Let be a sequence of positive numbers such that
The sequence-to-sequence transformation
defines the sequence of the mean of the sequence , generated by the sequence of coefficients (see ). The series is said to be summable , , if (see )
Let be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence to , where
The series is said to be summable , , if (see )
Let be any sequence of positive real numbers. The series is said to be summable , , if (see )
If we take , then summability reduces to summability. Also, if we take and , then we get summability. Furthermore, if we take , and for all values of n, reduces to summability. Finally, if we take and , then we get summability (see ).
Before stating the main theorem, we must first introduce some further notations.
Given a normal matrix , we associate two lower semimatrices and as follows:
It may be noted that and are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then we have
2 Known result
Many works have been done dealing with summability factors of infinite series (see [7–22]). Among them, in , the following main theorem has been proved.
Theorem A Let be a positive non-decreasing sequence and let there be sequences and such that
are satisfied. Furthermore, if is a sequence of positive numbers such that
then the series is summable , .
3 The main result
The aim of this paper is to generalize Theorem A for summability under weaker conditions. For this, we need the concept of an almost increasing sequence. A positive sequence is said to be almost increasing if there exists a positive increasing sequence and two positive constants A and B such that (see ). Obviously, every increasing sequence is an almost increasing sequence but the converse need not be true as can be seen from the example . Also, one can find some results dealing with absolute almost convergent sequences (see ). So, we are weakening the hypotheses of Theorem A replacing the increasing sequence by an almost increasing sequence. Now, we shall prove the following theorem.
Theorem Let be a positive normal matrix such that
Let be an almost increasing sequence and be a non-increasing sequence. If conditions (12)-(16) and
are satisfied, then the series is summable , .
Remark It should be noted that if we take as a positive non-decreasing sequence, and , then we get Theorem A. In this case, conditions (21) and (22) reduce to conditions (16) and (17), respectively. Also, the condition ‘ is a non-increasing sequence’ and the conditions (18)-(20) are automatically satisfied.
Under the conditions on , and as taken in the statement of the theorem, we have the following:
Proof of the Theorem Let denote A-transform of the series . Then we have, by (10) and (11),
Applying Abel’s transformation to this sum, we get that
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
Now, when , applying Hölder’s inequality with indices k and , where , we have that
by virtue of the hypotheses of the theorem and the lemma. Again, applying Hölder’s inequality and using the fact that by (23), we get that
by virtue of the hypotheses of the theorem and the lemma. Finally, as in , we have that
This completes the proof of the theorem. If we take , then we get a result concerning the summability factors. If we take , then we have another result dealing with summability. If we take and for all values of n, then we get a result dealing with summability. If we take , and for all values of n, then we get a result for summability. □
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The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Özarslan, H., Keten, A. On a new application of almost increasing sequences. J Inequal Appl 2013, 13 (2013). https://doi.org/10.1186/1029-242X-2013-13
- absolute matrix summability
- almost increasing sequences
- infinite series