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In the present paper, we obtained a main theorem related to factored infinite series. Some new results are also deduced.
Let ∑ an be a given infinite series with (sn) as the sequence of partial sums. In , Borwein introduced the (C, α, β) methods in the following form: Let α + β ≠ −1, −2, …. Then the (C, α, β) mean is defined by
The series ∑ an is said to be summable |C, α, β, σ; δ|k, k ≥ 1, δ ≥ 0, α + β > −1, and σ ∈ R, if (see )
Here, we shall prove the following theorem.
is a convex sequence (see ) such that the series
holds, then the series ∑ anλn is summable |C, α, β, σ; δ|k, k ≥ 1, 0 ≤ δ < α ≤ 1, σ ∈ R, and (α + β + 1)k − σ(δk + k − 1) > 1.
One should note that, if we set σ = 1, then we obtain a well-known result of Bor (see ).
We will use the following lemmas for the proof of the theorem given above.
If 0 < α ≤ 1, β > −1, and 1 ≤ v ≤ n, then
is a convex sequence such that the series
First applying Abel’s transformation and then using Lemma 1, we have
In order to complete the proof of the theorem by using Minkowski’s inequality, it is sufficient to show that
For k > 1, we can apply Hölder’s inequality with indices k and k′, where
by virtue of hypotheses of the theorem and Lemma 2. Similarly, we have
in view of hypotheses of the theorem and Lemma 2. This completes the proof of the theorem.
By selecting proper values for α, β, δ, and σ, we have some new results concerning the |C, 1|k, |C, α|k, and |C, α; δ|k summability methods.
The author declares that he has no competing interests.
The author carried out all work of this article and the main theorem. The author read and approved the final manuscript.
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