Abstract
New results concerning product summability of an infinite series are given. Some special cases are also deduced.
1. Introduction
Let be a given infinite series with partial sums . Let denote the nth Cesaro mean of order of the sequence . The series is summable if
(Flett [1]). For reduces to summability.
Let be a sequence of positive real constants such that as The transform of generated by is defined by
The sequence-to-sequence transformation defines the sequence of transform of generated by The series is summable if
In the special case when for all n (resp., k = 1), summability reduces to (resp., ) summability.
The series is said to be summable , when the transform of the transform of is a sequence of bounded variation (see Das [2]).
We give the following new definition.
Let define the sequence of the transform of the transform of generated by the sequences and respectively. The series is said to be summable if
We may assume through the paper that as ; as .
2. New Results
We state and prove the following.
Theorem 2.1. Let be a sequence of constants. Define
Let Then, sufficient conditions for the implication
are
Proof. Let be the sequence of partial sums of . Let be the , transforms of the sequences respectively. We write Therefore,
Also,
In order to complete the proof, it is sufficient to show that
Applying Holder's inequality,
Finally,
This completes the
proof of the theorem.
Theorem 2.2. Let (2.3) be
satisfied and
Then, necessary conditions for the implication (2.4) to be satisfied are
Proof. For define
From (2.14), we have
With and as defined by (2.12) and (2.22), the spaces and are BK-spaces with norms defined by
respectively. By the hypothesis
of the theorem,
The inclusion map defined by is continuous since and are BK-spaces. By the closed graph
theorem, there exists
a constant such that
Let denote the nth coordinate vector. From (2.12) and (2.22) with defined by otherwise, we have
From (2.23), we have
Applying (2.25), we obtain
As the right-hand side of (2.28), by (2.3), is
and the fact that each term of
the left-hand side of (2.28) is , we obtain
which implies by (2.18)
that is,
Also, we have, by (2.28),
The above, via the linear independence of and , implies
by (2.18). As by (2.19), via the mean value theorem,
Then,
which implies
Also, by (2.28),
which implies
3. Applications
Corollary 3.1. Let Define
Let
Then, sufficient conditions for
the implication
are (2.5), (2.6), and the following:
Proof. The proof follows from Theorem 2.1 by putting for all n.
Corollary 3.2. Let Define
Let (2.2) be satisfied. Then, sufficient conditions for the implication
are
Proof. The proof follows from Theorem
2.1, by putting for all n, noticing that (2.3) is satisfied as
Corollary 3.3. Let be as defined in (3.1). Let (2.3) and (3.2) be satisfied. Then, sufficient conditions for
the implication
are (2.5), (2.6), (2.10), (2.11), and the following:
Proof. The proof follows from Theorem 2.1, by outing for all n.
Corollary 3.4. Let be as defined in (3.1). Let (2.3), (2.19) be satisfied and
Then, necessary conditions for
the implication (3.3) are
Proof. The proof follows from Theorem 2.2 by putting for all n.
Corollary 3.5. Let be as defined in (3.5). Let
Then, necessary conditions for
the implication (3.5) to be satisfied are
Proof. The proof follows from Theorem 2.2, by putting keeping in mind that (2.3) is satisfied as in the case of (3.8).
Corollary 3.6. Let be as defined in (3.1). Let (2.3), (2.19), and (3.2) be all
satisfied. Then, necessary conditions
for the implication (3.9) to be satisfied
are
Proof. The proof follows from Theorem 2.2, by putting for all n.