In a recent paper [Savaş and Rhoades in Appl. Math. Lett. 22:1462-1466, 2009], the authors extended the result of Flett [Proc. Lond. Math. Soc. 7:113-141, 1957] to double summability. In this paper, we consider some further extensions of absolute Cesàro summability for double series.
MSC:40F05, 40G05.
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
for .
A four-dimensional matrix is said to be absolutely k th power conservative for , if , i.e., if
then
where
A double infinite Cesáro matrix is a double infinite Hausdorff matrix with entries
where
The series is said to be summable , , , if (see [1])
(1)
where denotes the mn-term of the transform of a sequence ; i.e.,
(2)
Quite recently, Savaş and Rhoades [2] extended the result of Flett [3] to double summability. Their theorem is as follows.
Theorem 1Let , , andbe a double series with partial sums . Ifis -summable, then it is also -summable, .
It then follows that if one sets , then for each . In this paper, we consider some further extensions of absolute Cesàro summability for double series.
(iv) Ifand the conditionis satisfied thenfor each .
(v) If the conditionis satisfied thenfor eachand .
(vi) Ifand the conditionis satisfied thenfor each .
(vii) If the conditionis satisfied thenfor each , .
(viii) If the conditionis satisfied thenfor eachand .
(ix) Ifand the conditionis satisfied thenfor each .
Proof We shall show that , i.e.,
Let denote the mn-term of the -transform in terms of , i.e.,
For , since
to prove the theorem, it will be sufficient to show that
(7)
Using Hölder’s inequality, we have
Since
and using the fact that
we obtain
Applying Hölder’s inequality with indices , , we deduce that
Since , we have
(i) From Lemma 1, if , then
Therefore, for the case , we have
since .
(ii) If , from Lemma 2, then
Hence,
(iii) If , , from Lemma 2, then
and then
(iv) If and , from Lemma 2, then
therefore, we have
(v) If and , then
by using Lemma 2.
(vi) If and , then
by using Lemma 2.
(vii) If and , then
by using Lemma 2.
(viii) If and , then
by using Lemma 2.
(ix) If and , then
by using Lemma 2. □
The one-dimensional version of Theorem 2 appears in [6]. By (5), Theorem 2 includes the following theorem with the special case .
Theorem 3Let .
(i) It holds thatfor each .
(ii) If the conditionis satisfied thenfor eachand .
(iii) If the conditionis satisfied thenfor eachand .
(iv) If the conditionis satisfied thenfor eachand .
Remark 1 Theorem 3 moderates Theorem 1 of [7]. Since Holder’s inequality is valid if each of the terms is nonnegative, it should be added the absolute values of the binomial coefficients in the proof of Theorem 1 of [7], when and/or . Therefore, if we replace the binomial coefficients with their absolute values, then the inequality (15) of [7] will be true. So, we should add the conditions, given above in (ii), (iii) and (iv) of Theorem 3 in the statement of Theorem 1 of [7], for the cases and/or .
Corollary 1Let .
(i) It holds thatfor each .
(ii) If the conditionis satisfied thenfor each .
Corollary 2Let .
(i) It holds thatfor each .
(ii) If the conditionis satisfied thenfor each .
Corollary 3Let . Then .
References
Rhoades BE: Absolute comparison theorems for double weighted mean and double Cesàro means. Math. Slovaca 1998, 48: 285–301.
Savaş E, Rhoades BE: An inclusion theorem for double Cesàro matrices over the space of absolutely k -convergent double series. Appl. Math. Lett. 2009, 22: 1462–1466. 10.1016/j.aml.2009.03.015
Flett TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math. Soc. 1957, 7: 113–141. 10.1112/plms/s3-7.1.113
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. Both the authors read and approved the final manuscript.
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Şevli, H., Savaş, E. Some further extensions of absolute Cesàro summability for double series.
J Inequal Appl2013, 144 (2013). https://doi.org/10.1186/1029-242X-2013-144