Some New Results on Absolute Summability Factors

Doi:10.18466/cbayarfbe.619883 G. C. Hazar Güleç Celal Bayar University Journal of Science

Some New Results on Absolute Summability Factors

G. Canan HAZAR GÜLEǹ*

1Department of Mathematics, Pamukkale University, 20070, Denizli.

*gchazar@pau.edu.tr Received: 13 September 2019

Accepted: 17 March 2020 DOI: 10.18466/cbayarfbe.619883

1. Introduction

Let ∑𝑥_𝑛 be a given infinite series with sequence of partial sums (𝑠_𝑛) and 𝐴 = (𝑎_𝑛𝑣) be an infinite matrix of complex numbers. By 𝐴(𝑠) = (𝐴_𝑛(𝑠)), we denote the 𝐴-transform of the sequence 𝑠 = (𝑠𝑛), i.e.,

𝐴_𝑛(𝑠) = ∑ 𝑎_𝑛𝑣𝑠_𝑣

∞

𝑣=0

which converges for 𝑛 ≥ 0.

The 𝑛th (𝑁̅, 𝑝𝑛) weighted mean of the sequence (𝑠_𝑛) is given by

𝑇_𝑛= 1 𝑃𝑛

∑ 𝑝_𝑣𝑠_𝑣

𝑛

𝑣=0

,

where (𝑝𝑛) is a sequence of positive real constants such that 𝑃𝑛= ∑𝑛 𝑝𝑣

𝑣=0 → ∞ as 𝑛 → ∞ (𝑃−1= 𝑝−1= 0).

Let (𝜑_𝑛) be any sequence of positive real constants.

Then the series ∑𝑥𝑛 is said to be summable

|𝑁̅, 𝑝𝑛, 𝜑𝑛|_𝑘, 𝑘 ≥ 1, if (see [1])

∑(𝜑𝑛)^𝑘−1

∞

𝑛=1

|𝑇_𝑛− 𝑇𝑛−1|^𝑘 < ∞. (1.1)

Note that |𝑁̅, 𝑝𝑛, 𝑃𝑛⁄𝑝𝑛|_𝑘= |𝑁̅, 𝑝𝑛|_𝑘 and |𝑁̅, 𝑝𝑛, 𝑛|𝑘 =

|𝑅, 𝑝_𝑛|_𝑘, which are defined by Bor and Sarıgöl in [2,3].

Taking account of

𝑇_𝑛− 𝑇_𝑛−1= 𝑝𝑛

𝑃_𝑛𝑃_𝑛−1∑ 𝑃_𝑣−1𝑥_𝑣

𝑛

𝑣=1

the relation (1.1) can be stated as

∑(𝜑_𝑛)^𝑘−1

∞

𝑛=1

| 𝑝𝑛

𝑃𝑛𝑃𝑛−1

∑ 𝑃_𝑣−1𝑥_𝑣

𝑛

𝑣=1

|

𝑘

< ∞. (1.2)

An appropriate extension of (1.2) to a factorable matrix would be as follows [4]. Let 𝐴_𝑓 = (𝑎_𝑛𝑣) denote the factorable matrix defined by

𝑎𝑛𝑣= {𝑎̂𝑛𝑎𝑣, 0 ≤ 𝑣 ≤ 𝑛, 0, 𝑣 > 𝑛,

where (𝑎̂_𝑛) and (𝑎_𝑛) are any sequences of real numbers.

Then the series ∑𝑥_𝑛 is said to be summable |𝐴_𝑓, 𝜑_𝑛|

𝑘, 𝑘 ≥ 1, if (see [4])

∑(𝜑_𝑛)^𝑘−1

∞

𝑛=1

|𝑎̂_𝑛∑ 𝑎_𝑣𝑥_𝑣

𝑛

𝑣=1

|

𝑘

< ∞.

If we take 𝑎̂𝑛= ^𝑝𝑛

𝑃_𝑛𝑃_𝑛−1 and 𝑎𝑣= 𝑃𝑣−1, then |𝐴𝑓, 𝜑𝑛|

𝑘

summability is equivalent to |𝑁̅, 𝑝𝑛, 𝜑𝑛|_𝑘 summability.

Abstract

In this paper, we establish the general summability factor theorems related to generalized absolute Cesàro summability |𝐶, 𝛼, 𝛽|𝑘 and absolute factorable matrix summability |𝐴_𝑓, 𝜑_𝑛|

𝑘 methods for 𝑘 ≥ 1, 𝛼 + 𝛽 > −1, where (𝜑_𝑛) is arbitrary sequence of positive real constants and 𝐴_𝑓= (𝑎_𝑛𝑣) is a factorable matrix such that 𝑎_𝑛𝑣= 𝑎̂_𝑛𝑎_𝑣 for 0 ≤ 𝑣 ≤ 𝑛, 𝑎_𝑛𝑣= 0 for 𝑣 > 𝑛, (𝑎̂_𝑛) and (𝑎_𝑛) are any sequences of real numbers. Also, absolute factorable summability method includes all absolute Riesz summability and absolute weighted summability methods in the special cases. Therefore, not only some well known results but also several new results for absolute Cesàro and weighted means are obtained as corollaries.

Keywords: Absolute Cesàro summability, Factorable matrix, Matrix methods, Sequence spaces, Summability factors.

Borwein [5] has introduced the 𝑛th generalized Cesàro mean (𝐶, 𝛼, 𝛽) of order (𝛼, 𝛽) with 𝛼 + 𝛽 > −1, of the sequence (𝑠𝑛) by

𝜎_𝑛^𝛼,𝛽= 1

𝐴_𝑛^𝛼+𝛽∑ 𝐴_𝑛−𝑣^𝛼−1

𝑛

𝑣=1

𝐴_𝑣^𝛽𝑠_𝑣,

where 𝐴_𝑛^𝛼+𝛽= 𝑂(𝑛^𝛼+𝛽), 𝛼 + 𝛽 > −1, 𝐴₀^𝛼+𝛽= 1, 𝐴𝑛𝛼=(𝛼+1)(𝛼+2)…(𝛼+𝑛)

𝑛! 𝑎𝑛𝑑 𝐴_−𝑛^𝛼+𝛽= 0, 𝑛 ≥ 1.

Obviously, (𝐶, 𝛼, 0) is the same as (𝐶, 𝛼) whereas (𝐶, 0, 𝛽) is (𝐶, 0).

We write 𝜏_𝑛^𝛼,𝛽 as the (𝐶, 𝛼, 𝛽) transform of the sequence (𝑛𝑥_𝑛), i.e.,

𝜏_𝑛^𝛼,𝛽= 1

𝐴^𝛼+𝛽_𝑛 ∑ 𝐴_𝑛−𝑣^𝛼−1

𝑛

𝑣=1

𝐴^𝛽_𝑣𝑣𝑥_𝑣.

Then, the series ∑𝑥_𝑛 is said to be summable |𝐶, 𝛼, 𝛽|_𝑘, 𝑘 ≥ 1, for 𝛼 + 𝛽 > −1, if (see [6])

∑1

𝑛

∞

𝑛=1

|𝜏_𝑛^𝛼,𝛽|^𝑘< ∞.

The summability |𝐶, 𝛼, 𝛽|_𝑘 includes all Cesàro methods in the special cases. For example, if we take 𝛽 = 0, 𝛼 = 0 and 𝛼 = 1, then the summability |𝐶, 𝛼, 𝛽|𝑘

reduces to |𝐶, 𝛼|_𝑘 defined by Flett in [7], to |𝐶, 0|_𝑘 and the absolute Riesz summability |𝑅, 𝑝_𝑛|_𝑘 with 𝑝_𝑛= 𝐴_𝑛^𝛽 for 𝛽 ≥ 0 [3].

Throughout this paper, 𝑘^∗ denotes the conjugate of 𝑘 > 1, i.e., 1/𝑘 + 1/ 𝑘^∗= 1, and 1/ 𝑘^∗= 0 for 𝑘 = 1.

Let 𝑋 and 𝑌 be two summability methods. If ∑ 𝜀𝑛𝑥𝑛 is summable by the method 𝑌 whenever ∑ 𝑥_𝑛 is summable by the method 𝑋, then we say that the sequence 𝜀 = (𝜀_𝑛) is a summability factor of type (𝑋, 𝑌) and we write 𝜀 ∈ (𝑋, 𝑌). Also, note that if 𝜀 = 1, then 1 ∈ (𝑋, 𝑌) means the comparisons of these methods, where 1 = (1,1, . . . ) , i.e., 𝑋 ⊂ 𝑌.

Absolute summability factors and comparison of the methods related to |𝑁̅, 𝑝_𝑛|_𝑘 and |𝐶, 𝛼|_𝑘 were widely studied by many authors [8-12]. We refer the reader to [11-13] for the most recent work in this topic. Also the Cesàro series spaces have been defined as the set of all series summable by absolute Cesàro summability methods in [14-16].

2. Results and Discussion

The aim of this paper is to characterize the sets (|𝐶, 𝛼, 𝛽|, |𝐴𝑓, 𝜑𝑛|

𝑘), 𝑘 ≥ 1 and

(|𝐴 , 𝜑 | , |𝐶, 𝛼, 𝛽|) , 𝑘 > 1 for 𝛼 + 𝛽 > −1. As a

direct consequence of these results, we also obtain various new results as corollaries.

We use the following lemmas to prove our results.

Lemma 2.1. Let 1 < 𝑘 < ∞. Then, 𝐴(𝑥) ∈ ℓ whenever 𝑥 ∈ ℓ𝑘 if and only if

∑ (∑|𝑎_𝑛𝑣|

∞

𝑛=0

)

𝑘^∗

∞

𝑣=0

< ∞

where ℓ_𝑘 = {𝑥 = (𝑥_𝑣) ∶ ∑ |𝑥_{𝑣 𝑣}|^𝑘 < ∞}, ℓ₁= ℓ, [17].

Lemma 2.2. Let 1 ≤ 𝑘 < ∞. Then, 𝐴(𝑥) ∈ ℓ_𝑘 whenever 𝑥 ∈ ℓ if and only if

sup

𝑣 ∑|𝑎𝑛𝑣|^𝑘

∞

𝑛=0

< ∞, [18].

Lemma 2.3. Let 𝜇 > −1, 1 ≤ 𝑘 < ∞ and 𝜆 < 𝜇. Then, for 𝑘 = 1,

𝐸_𝑣= {𝑂(𝑣^−𝜇−1), 𝜆 ≤ −1 𝑂(𝑣^−𝜇+𝜆), 𝜆 > −1 and

𝐸𝑣= {

𝑂(𝑣^{−𝑘𝜇−1}), 𝜆 < −1 𝑘⁄ 𝑂(𝑣^{−𝑘𝜇−1}𝑙𝑜𝑔𝑣), 𝜆 = −1 𝑘⁄

𝑂(𝑣^{−𝑘𝜇+𝑘𝜆}), 𝜆 > −1 𝑘⁄

for 1 < 𝑘 < ∞, where 𝐸𝑣 = ∑ ^{|𝐴𝑛−𝑣𝜆 |}

𝑘 𝑛(𝐴_𝑛^𝜇)^𝑘

∞𝑛=𝑣 for 𝑣 ≥ 1, [9].

Now, we are ready to prove the main theorems.

Theorem 2.4. Let 𝑘 ≥ 1 and 𝛼 + 𝛽 > −1. Then the necessary and sufficient condition for 𝜀 ∈ (|𝐶, 𝛼, 𝛽|, |𝐴𝑓, 𝜑𝑛|

𝑘) is that

sup

𝑟

{∑ |𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂_𝑛𝑟𝐴_𝑟^𝛼+𝛽∑𝑎_𝑣𝜀_𝑣𝐴_𝑣−𝑟^−𝛼−1 𝑣𝐴_𝑣^𝛽

𝑛

𝑣=𝑟

|

∞ 𝑘

𝑛=𝑟

} < ∞. (2.1)

Proof. Let 𝜏_𝑛^𝛼,𝛽 be the 𝑛th (𝐶, 𝛼, 𝛽) mean of the sequence (𝑛𝑥𝑛) and define the sequence (𝑦_𝑛) by

𝑦_𝑛=𝜏_𝑛^𝛼,𝛽

𝑛 = 1

𝑛𝐴_𝑛^𝛼+𝛽∑ 𝐴_𝑛−𝑣^𝛼−1

𝑛

𝑣=1

𝐴_𝑣^𝛽𝑣𝑥_𝑣,

𝑛 ≥ 1 𝑎𝑛𝑑 𝑦0= 𝑥0. (2.2) So, ∑𝑥_𝑛 is summable |𝐶, 𝛼, 𝛽| iff 𝑦 = (𝑦_𝑛) ∈ ℓ. Also, by inversion of (2.2), we have for 𝑛 ≥ 1

𝑥_𝑛= 1

𝑛𝐴_𝑛^𝛽∑ 𝐴_𝑛−𝑣^−𝛼−1𝑣𝐴_𝑣^𝛼+𝛽𝑦_𝑣.

𝑛

𝑣=1

(2.3)

Doi:10.18466/cbayarfbe.619883 G. C. Hazar Güleç Using definition of factorable matrix 𝐴𝑓, we define the

sequence (𝑦̃𝑛) by

𝑦̃𝑛= 𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂𝑛∑ 𝑎𝑣𝑥𝑣𝜀𝑣 𝑛

𝑣=1

, 𝑦̃0= 𝜀0𝑥0.

This gives us that ∑𝜀𝑛𝑥𝑛 is summable |𝐴𝑓, 𝜑𝑛|

𝑘 iff 𝑦̃ = (𝑦̃_𝑛) ∈ ℓ_𝑘.

Hence, in view of (2.3), we get for 𝑛 ≥ 1, 𝑦̃_𝑛= 𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂_𝑛∑ 𝑎_𝑣𝜀_𝑣

𝑛

𝑣=1

𝑥_𝑣

= 𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂_𝑛∑ 𝑎_𝑣𝜀_𝑣

𝑛

𝑣=1

1

𝑣𝐴_𝑣^𝛽∑ 𝐴_𝑣−𝑟^−𝛼−1𝑟𝐴_𝑟^𝛼+𝛽𝑦_𝑟

𝑣

𝑟=1

= 𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂_𝑛∑

𝑛

𝑟=1

(𝑟𝐴_𝑟^𝛼+𝛽∑𝑎𝑣𝜀𝑣𝐴𝑣−𝑟−𝛼−1

𝑣𝐴^𝛽_𝑣

𝑛

𝑣=𝑟

) 𝑦_𝑟

= ∑ 𝑑_𝑛𝑟

𝑛

𝑟=1

𝑦_𝑟 where

𝑑_𝑛𝑟= {𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂𝑛𝑟𝐴_𝑟^𝛼+𝛽∑𝑎_𝑣𝜀_𝑣𝐴_𝑣−𝑟^−𝛼−1 𝑣𝐴_𝑣^𝛽

𝑛

𝑣=𝑟

, 1 ≤ 𝑟 ≤ 𝑛 0, 𝑟 > 𝑛.

Then, ∑𝜀_𝑛𝑥_𝑛 is summable |𝐴_𝑓, 𝜑_𝑛|

𝑘 whenever ∑𝑥_𝑛 is summable |𝐶, 𝛼, 𝛽| if and only if 𝑦̃ ∈ ℓ𝑘 whenever 𝑦 ∈ ℓ. Hence using Lemma 2.2, we obtain that 𝜀 ∈ (|𝐶, 𝛼, 𝛽|, |𝐴𝑓, 𝜑𝑛|

𝑘) if and only if sup

𝑟

{∑ |𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂𝑛𝑟𝐴_𝑟^𝛼+𝛽∑𝑎_𝑣𝜀_𝑣𝐴_𝑣−𝑟^−𝛼−1 𝑣𝐴_𝑣^𝛽

𝑛

𝑣=𝑟

|

∞ 𝑘

𝑛=𝑟

} < ∞

which completes the proof.

Theorem 2.5. Let 𝑘 > 1, 𝛼 + 𝛽 > −1 and 𝛽 > −1.

Then the necessary and sufficient condition for 𝜀 ∈ (|𝐴𝑓, 𝜑𝑛|

𝑘 , |𝐶, 𝛼, 𝛽|) is that

∑ (∑ | 1

𝑛𝐴^𝛼+𝛽_𝑛 𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂𝑣

Ω_𝑛𝑣|

∞

𝑛=𝑣

)

𝑘^∗

∞

𝑣=1

< ∞, (2.4)

where Ω = (Ω_𝑛𝑣) is defined by

Ω_𝑛𝑣= {

𝐴_𝑛−𝑣^𝛼−1𝐴^𝛽_𝑣𝑣𝜀_𝑣

𝑎𝑣 −𝐴^𝛼−1_{𝑛−𝑣−1}𝐴_𝑣+1^𝛽 (𝑣+1)𝜀_𝑣+1

𝑎𝑣+1 , 1 ≤ 𝑣 ≤ 𝑛, 0, 𝑣 > 𝑛.

Proof. Let (𝑦̃_𝑛) denote the sequence defined by

𝑦̃_𝑛= 𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂_𝑛∑ 𝑎_𝑣𝑥_𝑣

𝑛

𝑣=1

, 𝑛 ≥ 1, 𝑎𝑛𝑑 𝑦̃₀= 𝑥₀. (2.5)

So, we can write that ∑𝑥𝑛 is summable |𝐴𝑓, 𝜑𝑛|

𝑘 iff 𝑦̃ = (𝑦̃𝑛) ∈ ℓ_𝑘. By inversion of (2.5), we obtain for 𝑛 ≥ 1,

𝑥𝑛= 1 𝑎𝑛

( 𝑦̃_𝑛 𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂𝑛

− 𝑦̃_𝑛−1 𝜑_𝑛−1^{1 𝑘⁄ ∗}𝑎̂𝑛−1

). (2.6)

Also let (𝑢_𝑛^𝛼,𝛽) be the 𝑛th (𝐶, 𝛼, 𝛽) mean of the sequence (𝑛𝑥_𝑛𝜀_𝑛), i.e.,

𝑢_𝑛^𝛼,𝛽= 1

𝐴^𝛼+𝛽_𝑛 ∑ 𝐴𝑛−𝑣𝛼−1 𝑛

𝑣=1

𝐴_𝑣^𝛽𝑣𝜀𝑣𝑥𝑣.

If we define 𝑦 = (𝑦_𝑛) by

𝑦𝑛=𝑢_𝑛^𝛼,𝛽

𝑛 = 1

𝑛𝐴_𝑛^𝛼+𝛽∑ 𝐴𝛼−1𝑛−𝑣 𝑛

𝑣=1

𝐴_𝑣^𝛽𝑣𝜀𝑣𝑥𝑣,

then, we say that ∑𝜀_𝑛𝑥_𝑛 is summable |𝐶, 𝛼, 𝛽| iff 𝑦 = (𝑦𝑛) ∈ ℓ. Hence, by virtue of the (2.6), we get for 𝑛 ≥ 1,

𝑦_𝑛= 1

𝑛𝐴_𝑛^𝛼+𝛽∑ 𝐴_𝑛−𝑣^𝛼−1

𝑛

𝑣=1

𝐴_𝑣^𝛽𝑣𝜀_𝑣𝑥_𝑣

= 1

𝑛𝐴_𝑛^𝛼+𝛽∑ 𝐴_𝑛−𝑣^𝛼−1

𝑛

𝑣=1

𝐴_𝑣^𝛽𝑣𝜀_𝑣 1 𝑎_𝑣( 𝑦̃_𝑣

𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂_𝑣− 𝑦̃_𝑣−1 𝜑_𝑣−1^{1 𝑘⁄ ∗}𝑎̂_𝑣−1)

= 1

𝑛𝐴_𝑛^𝛼+𝛽(∑𝐴^𝛼−1_𝑛−𝑣𝐴_𝑣^𝛽𝑣𝜀_𝑣𝑦̃_𝑣 𝑎_𝑣𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂_𝑣

𝑛

𝑣=1

− ∑𝐴𝛼−1𝑛−𝑣−1𝐴_𝑣+1^𝛽 (𝑣 + 1)𝜀𝑣+1𝑦̃𝑣

𝑎_𝑣+1𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂_𝑣

𝑛−1

𝑣=0

)

= − 𝐴𝑛−1𝛼−1𝐴₁^𝛽𝜀1𝑦̃0

𝑛𝐴_𝑛^𝛼+𝛽𝑎1𝜑₀^{1 𝑘⁄ ∗}𝑎̂0

+ 1

𝑛𝐴_𝑛^𝛼+𝛽∑ (𝐴𝑛−𝑣𝛼−1𝐴_𝑣^𝛽𝑣𝜀𝑣

𝑎_𝑣

𝑛

𝑣=1

−𝐴𝑛−𝑣−1𝛼−1 𝐴_𝑣+1^𝛽 (𝑣 + 1)𝜀𝑣+1

𝑎_𝑣+1 ) 𝑦̃𝑣

𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂𝑣

= ∑ 𝑑_𝑛𝑣𝑦̃_𝑣

𝑛

𝑣=0

where 𝐷 = (𝑑𝑛𝑣) is defined by

𝑑𝑛𝑣=

{

− 𝐴_𝑛−1^𝛼−1𝐴₁^𝛽𝜀1

𝑛𝐴_𝑛^𝛼+𝛽𝑎1𝜑₀^{1 𝑘⁄ ∗}𝑎̂0

, 𝑣 = 0, 𝑛 ≥ 1, 1

𝑛𝐴_𝑛^𝛼+𝛽𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂_𝑣Ω𝑛𝑣 , 1 ≤ 𝑣 ≤ 𝑛 0, 𝑣 > 𝑛,

and Ω = (Ω_𝑛𝑣) is as in Theorem 2.5.

Then, ∑𝜀𝑛𝑥𝑛 is summable |𝐶, 𝛼, 𝛽| whenever ∑ 𝑥𝑛 is summable |𝐴𝑓, 𝜑𝑛|

𝑘 if and only if 𝑦 ∈ ℓ whenever 𝑦̃ ∈ ℓ_𝑘. Hence in view of Lemma 2.1, we obtain that 𝜀 ∈ (|𝐴_𝑓, 𝜑_𝑛|

𝑘 , |𝐶, 𝛼, 𝛽|) if and only if

∑ (∑|𝑑𝑛𝑣|

∞

𝑛=𝑣

)

𝑘^∗

∞

𝑣=0

< ∞

which gives that

(∑|𝑑_𝑛0|

∞

𝑛=1

)

𝑘^∗

+ ∑ (∑|𝑑_𝑛𝑣|

∞

𝑛=𝑣

)

𝑘^∗

∞

𝑣=1

= (∑ | 𝐴_𝑛−1^𝛼−1𝐴₁^𝛽𝜀1

𝑛𝐴_𝑛^𝛼+𝛽𝑎1𝜑₀^{1 𝑘⁄ ∗}𝑎̂0

|

∞

𝑛=1

)

𝑘^∗

+ ∑ (∑ | 1

𝑛𝐴^𝛼+𝛽_𝑛 𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂_𝑣(𝐴^𝛼−1_𝑛−𝑣𝐴_𝑣^𝛽𝑣𝜀_𝑣 𝑎𝑣

∞

𝑛=𝑣

∞

𝑣=1

−𝐴_{𝑛−𝑣−1}^𝛼−1 𝐴_𝑣+1^𝛽 (𝑣 + 1)𝜀_𝑣+1 𝑎𝑣+1

)|)

𝑘^∗

< ∞.

Since ∑ |^{𝐴𝑛−1𝛼−1}

𝑛𝐴_𝑛^𝛼+𝛽|

∞𝑛=1 < ∞ from Lemma 2.3 , we get that (2.4) holds, which completes the proof.

3. Conclusion

Our results have several consequences depending on 𝛼, 𝛽, (𝑎̂_𝑛) and (𝑎_𝑛) .

If we consider the special case 𝜀 = 1 in the Theorem 2.4 and Theorem 2.5, we have following results dealing with comparison of summability fields of methods

|𝐶, 𝛼, 𝛽| and |𝐴_𝑓, 𝜑𝑛|

𝑘.

Corollary 3.1. Let 𝑘 ≥ 1 and 𝛼 + 𝛽 > −1. Then,

|𝐶, 𝛼, 𝛽| ⊂ |𝐴_𝑓, 𝜑𝑛|

𝑘 if and only if

sup

𝑟 {∑ |𝜑𝑛1 𝑘⁄ ^∗

𝑎̂𝑛𝑟𝐴^𝛼+𝛽_𝑟 ∑𝑎_𝑣𝐴_𝑣−𝑟^−𝛼−1 𝑣𝐴_𝑣^𝛽

𝑛

𝑣=𝑟

|

∞ 𝑘

𝑛=𝑟

} < ∞.

Corollary 3.2. Let 𝑘 > 1, 𝛼 + 𝛽 > −1 and 𝛽 > −1.

Then |𝐴_𝑓, 𝜑_𝑛|

𝑘⊂ |𝐶, 𝛼, 𝛽| if and only if

∑ (∑ | 1

𝑛𝐴_𝑛^𝛼+𝛽𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂_𝑣(𝐴_𝑛−𝑣^𝛼−1𝐴_𝑣^𝛽𝑣 𝑎_𝑣

∞

𝑛=𝑣

∞

𝑣=1

−𝐴^𝛼−1_{𝑛−𝑣−1}𝐴_𝑣+1^𝛽 (𝑣 + 1) 𝑎𝑣+1

)|)

𝑘^∗

< ∞.

Taking 𝑎̂𝑛= ^𝑝𝑛

𝑃_𝑛𝑃_𝑛−1 , 𝑎𝑣= 𝑃𝑣−1 in the Theorem 2.4 and Theorem 2.5, we get the following results, respectively.

Corollary 3.3. Let 𝑘 ≥ 1 and 𝛼 + 𝛽 > −1. Then the necessary and sufficient condition for 𝜀 ∈ (|𝐶, 𝛼, 𝛽|, |𝑁̅, 𝑝_𝑛, 𝜑_𝑛|_𝑘) is that

sup

𝑟

{∑ |𝜑_𝑛^{1 𝑘⁄ ∗} 𝑝_𝑛 𝑃𝑛𝑃𝑛−1

𝑟𝐴_𝑟^𝛼+𝛽∑𝑃_𝑣−1𝜀_𝑣𝐴_𝑣−𝑟^−𝛼−1 𝑣𝐴_𝑣^𝛽

𝑛

𝑣=𝑟

|

∞ 𝑘

𝑛=𝑟

}

< ∞.

Corollary 3.4. Let 𝑘 > 1, 𝛼 + 𝛽 > −1 and 𝛽 > −1.

Then the necessary and sufficient condition for 𝜀 ∈ (|𝑁̅, 𝑝𝑛, 𝜑_𝑛|_𝑘, |𝐶, 𝛼, 𝛽|) is that

∑ (∑ | 𝑃_𝑣𝑃_𝑣−1

𝑛𝐴^𝛼+𝛽_𝑛 𝜑_𝑣^{1 𝑘⁄ ∗}𝑝_𝑣(𝐴_𝑛−𝑣^𝛼−1𝐴_𝑣^𝛽𝑣𝜀_𝑣 𝑃𝑣−1

∞

𝑛=𝑣

∞

𝑣=1

−𝐴_{𝑛−𝑣−1}^𝛼−1 𝐴_𝑣+1^𝛽 (𝑣+1)𝜀𝑣+1

𝑃_𝑣 )|)

𝑘^∗

< ∞.

If we take 𝛽 = 0, Theorem 2.4 and Theorem 2.5 reduce to the next results, respectively.

Corollary 3.5. Let 𝑘 ≥ 1 and 𝛼 > −1. Then the necessary and sufficient condition for 𝜀 ∈ (|𝐶, 𝛼|, |𝐴_𝑓, 𝜑_𝑛|

𝑘) is that

sup

𝑟

{∑ |𝜑_𝑛^{1 𝑘⁄ ∗}𝑎̂_𝑛𝑟𝐴_𝑟^𝛼∑𝑎_𝑣𝜀_𝑣𝐴_𝑣−𝑟^−𝛼−1 𝑣

𝑛

𝑣=𝑟

|

∞ 𝑘

𝑛=𝑟

} < ∞.

Corollary 3.6. Let 𝑘 > 1 and 𝛼 > −1. Then the necessary and sufficient condition for 𝜀 ∈ (|𝐴_𝑓, 𝜑_𝑛|

𝑘, |𝐶, 𝛼|) is that

∑ (∑ | 1

𝑛𝐴_𝑛^𝛼𝜑_𝑣^{1 𝑘⁄ ∗}𝑎̂_𝑣(𝐴𝑛−𝑣𝛼−1𝑣𝜀𝑣

𝑎_𝑣

∞

𝑛=𝑣

∞

𝑣=1

−𝐴^𝛼−1_{𝑛−𝑣−1}(𝑣+1)𝜀_𝑣+1 𝑎𝑣+1

)|)

𝑘^∗

< ∞.

Also, taking 𝑎̂𝑛= ^𝑝𝑛

𝑃_𝑛𝑃_𝑛−1 , 𝑎𝑣= 𝑃𝑣−1 in the Corollary 3.5. and Corollary 3.6, we have:

Doi:10.18466/cbayarfbe.619883 G. C. Hazar Güleç Corollary 3.7. Let 𝑘 ≥ 1 and 𝛼 > −1. Then the

necessary and sufficient condition for 𝜀 ∈ (|𝐶, 𝛼|, |𝑁̅, 𝑝𝑛, 𝜑𝑛|_𝑘) is that

sup

𝑟 {∑ |𝜑_𝑛^{1 𝑘⁄ ∗} 𝑝_𝑛 𝑃𝑛𝑃𝑛−1

𝑟𝐴𝑟𝛼∑𝑃_𝑣−1𝜀_𝑣𝐴_𝑣−𝑟^−𝛼−1 𝑣

𝑛

𝑣=𝑟

|

∞ 𝑘

𝑛=𝑟

} < ∞.

Corollary 3.8. Let 𝑘 > 1 and 𝛼 > −1. Then the necessary and sufficient condition for 𝜀 ∈ (|𝑁̅, 𝑝𝑛, 𝜑_𝑛|_𝑘, |𝐶, 𝛼|) is that

∑ (∑ | 𝑃_𝑣𝑃_𝑣−1

𝑛𝐴_𝑛^𝛼𝜑_𝑣^{1 𝑘⁄ ∗}𝑝_𝑣(𝐴^𝛼−1_𝑛−𝑣𝑣𝜀_𝑣 𝑃𝑣−1

∞

𝑛=𝑣

∞

𝑣=1

−𝐴_{𝑛−𝑣−1}^𝛼−1 (𝑣+1)𝜀𝑣+1

𝑃_𝑣 )|)

𝑘^∗

< ∞.

Acknowledgement

Authors may acknowledge technical assistance, the source of special materials, financial support and the auspices under which work was done. The names of funding organizations should be written in full.

Ethics

There are no ethical issues after the publication of this manuscript.

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