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On double Hausdorff summability method
Journal of Inequalities and Applications volume 2014, Article number: 240 (2014)
Das (Proc. Camb. Philos. Soc. 67:321-326, 1970) proved that every conservative Hausdorff matrix is absolutely k th power conservative. Savaş and Rhoades (Anal. Math. 35:249-256, 2009) proved the result of Das for double Hausdorff summability. In this paper we will consider the double Endl-Jakimovski (E-J) generalization and we will prove the corresponding result of Savaş and Şevli (J. Comput. Anal. Appl. 11:702-710, 2009) for double E-J generalized Hausdorff matrices.
Introduction and background
Let β be a real number, let be a real sequence, and let Δ be the forward difference operator defined by , . Then the infinite matrix is defined by
and the associated matrix method is called a generalized Hausdorff matrix and generalized Hausdorff method, respectively. The moment sequence is given by
where . The case corresponds to ordinary Hausdorff summability.
In a recent paper , the first author jointly with Savaş has extended the result of Das  to the E-J matrices; i.e., all conservative E-J matrices are absolutely k th power conservative for . Thereafter, Savaş and Rhoades  proved the result of Das  for double Hausdorff summability. In this paper we will consider double E-J generalization and we will prove the corresponding result of  for double E-J generalized Hausdorff matrices.
Let be an infinite double series with real or complex numbers, with partial sums
For any double sequence we shall define
Denote by the sequence space defined by
A four-dimensional matrix is said to be absolutely k th power conservative for , if ; i.e., if
A double Hausdorff matrix has entries
where is any real or complex sequence and
For double Hausdorff matrices, the necessary and sufficient condition for H to be conservative is the existence of a function such that
Theorem 1 
Let H be a conservative double Hausdorff matrix. Then .
Our purpose is to achieve the result established in  for double E-J Hausdorff summability.
The matrix , whose elements are defined by
is called a difference matrix, where α and β are real numbers.
Theorem 2 The difference matrix is its own inverse.
thus . For any double sequence
Let be a given sequence and be a diagonal matrix whose only non-zero entries are . The transformation matrix
is called a double E-J generalized Hausdorff matrix corresponding to the sequence .
Theorem 3 A matrix is a double E-J generalized Hausdorff matrix corresponding to the sequence if and only if its elements have the form
Proof Let be a double E-J Hausdorff matrix. Applying this to a double sequence we have
For double E-J Hausdorff matrices, the necessary and sufficient condition for to be conservative is the existence of a function such that
Theorem 4 Given a function , a bounded variation in the unit square, the corresponding double E-J Hausdorff transformation , of a sequence , may be defined by
Proof For and ,
Theorem 5 Let be a conservative double E-J Hausdorff matrix. Then , .
As tools to prove our result, we need to the following lemmas.
Lemma 1 
Let , and . Then
The following lemma is a double version of .
Lemma 2 For , , and
Proof of Theorem 5 Let be the double E-J transform of a double sequence ; i.e.,
We will demonstrate that
Then . For
Due to this (1) is equivalent to
For and define
It follows from the Hölder inequality that
From Lemma 2
and from Lemma 1
From Lemma of , if and are the transformation of and , respectively, then
A similar consequence can be proved for , see ; i.e.,
Since is conservative, is a moment sequence,
from Theorem 4. In view of (3) we can deduce that
Using Minkowski’s inequality we get
Therefore the proof of Theorem 5 is complete. □
Specially, if we take and in Theorem 5, we get Theorem 1 as a corollary.
The following is an example of a double E-J Hausdorff matrix.
A doubly infinite Cesàro matrix is a doubly infinite Hausdorff matrix with entries
We use the following to denote the corresponding E-J generalizations of the .
has moment sequence
For and ,
For the special case ,
is a double E-J Hausdorff matrix.
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This work is supported by Istanbul Commerce University Scientific Research Projects Coordination Unit.
The authors declare that they have no competing interests.
The authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.
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Şevli, H., Savaş, R. On double Hausdorff summability method. J Inequal Appl 2014, 240 (2014). https://doi.org/10.1186/1029-242X-2014-240
- absolute summability
- conservative matrix
- double series
- Hausdorff matrices