Quasimonotone and Almost Increasing Sequences and Their New Applications

Abstract

Recently, we have proved a main theorem dealing with the absolute Nörlund summability factors of infinite series by using 𝛿-quasimonotone sequences. In this paper, we prove that result under weaker conditions. A new result has also been obtained.

1. Introduction

A positive sequence (𝑏_𝑛) is said to be almost increasing if there exist a positive increasing sequence (𝑐_𝑛) and two positive constants 𝐴 and 𝐵 such that 𝐴𝑐_𝑛≤𝑏_𝑛≤𝐵𝑐_𝑛 (see [1]). Obviously, every increasing sequence is almost increasing. However, the converse need not be true as can be seen by taking an example, say 𝑏_𝑛=𝑛𝑒^(−1)𝑛. A sequence (𝑑_𝑛) is said to be 𝛿-quasimonotone if 𝑑_𝑛>0 ultimately and Δ𝑑_𝑛=𝑑_𝑛−𝑑_𝑛+1≥−𝛿_𝑛, where 𝛿=(𝛿_𝑛) is a sequence of positive numbers (see [2]). Let ∑𝑎_𝑛 be a given infinite series with the sequence of partial sums (𝑠_𝑛) and 𝑤_𝑛=𝑛𝑎_𝑛. By 𝑢^𝛼_𝑛 and 𝑡^𝛼_𝑛, we denote the 𝑛th Cesàro means of order 𝛼, with 𝛼>−1, of the sequences (𝑠_𝑛) and (𝑛𝑎_𝑛), respectively, that is, 𝑢^𝛼_𝑛=1𝐴^𝛼_𝑛^𝑛_𝑣=0𝐴^𝛼−1_𝑛−𝑣𝑠_𝑣,𝑡(1.1)_𝛼𝑛=1𝐴^𝛼_𝑛^𝑛_𝑣=1𝐴^𝛼−1_𝑛−𝑣𝑣𝑎_𝑣,(1.2) where 𝐴^𝛼_𝑛=𝑛=𝑛+𝛼(𝛼+1)(𝛼+2)⋯(𝛼+𝑛)𝑛𝑛!=𝑂^𝛼,𝐴^𝛼_−𝑛=0for𝑛>0.(1.3) The series ∑𝑎_𝑛 is said to be summable |𝐶,𝛼|_𝑘, 𝑘≥1, if (see [3]) ^∞_𝑛=1𝑛^𝑘−1𝑢^𝛼_𝑛−𝑢^𝛼_𝑛−1^𝑘=^∞_𝑛=11𝑛𝑡^𝛼_𝑛^𝑘<∞.(1.4) If we take 𝛼=1, then |𝐶,𝛼|_𝑘 summability reduces to |𝐶,1|_𝑘 summability.

Let (𝑝_𝑛) be a sequence of constants, real or complex, and let us write 𝑃_𝑛=𝑝₀+𝑝₁+𝑝₂+⋯+𝑝_𝑛≠0,(𝑛≥0).(1.5) The sequence-to-sequence transformation 𝑉_𝑛=1𝑃_𝑛^𝑛_𝑣=0𝑝_𝑛−𝑣𝑠_𝑣(1.6) defines the sequence (𝑉_𝑛) of the Nörlund mean of the sequence (𝑠_𝑛), generated by the sequence of coefficients (𝑝_𝑛). The series ∑𝑎_𝑛 is said to be summable |𝑁,𝑝_𝑛|_𝑘, 𝑘≥1, if (see [4]) ^∞_𝑛=1𝑛^𝑘−1𝑉_𝑛−𝑉_𝑛−1^𝑘<∞.(1.7) In the special case when 𝑝_𝑛=Γ(𝑛+𝛼)Γ(𝛼)Γ(𝑛+1),𝛼≥0,(1.8) the Nörlund mean reduces to the (𝐶,𝛼) mean and |𝑁,𝑝_𝑛|_𝑘 summability becomes |𝐶,𝛼|_𝑘 summability. For 𝑝_𝑛=1, we get the (𝐶,1) mean and then |𝑁,𝑝_𝑛|_𝑘 summability becomes |𝐶,1|_𝑘 summability. Also, if we take 𝑘=1, then we get |𝑁,𝑝_𝑛| summability. For any sequence (𝜆_𝑛), we write Δ𝜆_𝑛=𝜆_𝑛−𝜆_𝑛+1. Quite recently, in [5], we have proved the following theorem dealing with the absolute Nörlund summability factors of infinite series.

Theorem A. Let 𝑝₀>0, 𝑝_𝑛≥0, and (𝑝_𝑛) be a nonincreasing sequence. Let (𝑋_𝑛) be an almost increasing sequence such that |Δ𝑋_𝑛|=𝑂(𝑋_𝑛/𝑛) and (𝜆_𝑛) is a sequence such that 𝜆_𝑛𝑋_𝑛=𝑂(1)𝑎𝑠𝑛→∞.(1.9) Suppose also that there exists a sequence of numbers (𝐴_𝑛) such that it is 𝛿-quasimonotone with ∑𝑛𝛿_𝑛𝑋_𝑛<∞, ∑𝐴_𝑛𝑋_𝑛 is convergent, and |Δ𝜆_𝑛|≤|𝐴_𝑛| for all 𝑛. If the sequence (𝑤^𝛼_𝑛) defined by (see [6]) 𝑤^𝛼_𝑛=𝑡^𝛼_𝑛,𝛼=1,max_{1≤𝑣≤𝑛}𝑡^𝛼_𝑣,0<𝛼<1,(1.10) satisfies the condition ^𝑚_𝑛=1𝑤^𝛼_𝑛^𝑘𝑛=𝑂𝑋_𝑚𝑎𝑠𝑚→∞,(1.11) then the series ∑𝑎_𝑛𝑃_𝑛𝜆_𝑛(𝑛+1)⁻¹ is summable |𝑁,𝑝_𝑛|_𝑘, 𝑘≥1.

2. The Main Results

The aim of this paper is to prove Theorem A under weaker conditions. We will prove the following theorems.

Theorem 2.1. If the sequences (𝑋_𝑛), (𝐴_𝑛), and (𝜆_𝑛) are as in Theorem A and if conditions (1.9) and ^𝑚_𝑛=1𝑤^𝛼_𝑛^𝑘𝑛𝑋_𝑛^𝑘−1=𝑂𝑋_𝑚𝑎𝑠𝑚→∞(2.1) are satisfied, then the series ∑𝑎_𝑛𝜆_𝑛 is summable |𝐶,𝛼|_𝑘, 0<𝛼≤1 and 𝑘≥1.

Theorem 2.2. Let (𝑝_𝑛) be as in Theorem A. If the sequences (𝑋_𝑛), (𝐴_𝑛), and (𝜆_𝑛) are as in Theorem A and if conditions (1.9) and (2.1) are satisfied, then the series ∑𝑎_𝑛𝑃_𝑛𝜆_𝑛(𝑛+1)⁻¹ is summable |𝑁,𝑝_𝑛|_𝑘, 𝑘≥1.

Remark 2.3. The following sequences satisfy the conditions of the theorems: 𝛿_𝑛=1𝑛³,𝐴_𝑛=1𝑛²,𝜆_𝑛=1𝑛,𝑋_𝑛=𝑛^𝜖,0<𝜖<1.(2.2)

Remark 2.4. It should be noted that condition (2.1) is the same as condition (1.11) when 𝑘=1. When 𝑘>1, condition (2.1) is weaker than condition (1.11), but the converse is not true. In fact, if (1.11) is satisfied, then we get that ^𝑚_𝑛=1𝑤^𝛼_𝑛^𝑘𝑛𝑋_𝑛^𝑘−11=𝑂𝑋₁^𝑘−1𝑚_𝑛=1𝑤^𝛼_𝑛^𝑘𝑛=𝑂𝑋_𝑚.(2.3) To show that the converse is false when 𝑘>1, the following example is sufficient. We can take 𝑋_𝑛=𝑛^𝜖, 0<𝜖<1, and then construct a sequence (𝑎_𝑛) such that 𝑤^𝛼_𝑛^𝑘𝑛𝑋_𝑛^𝑘−1=𝑋_𝑛−𝑋_𝑛−1,(2.4) whence ^𝑚_𝑛=1𝑤^𝛼_𝑛^𝑘𝑛𝑋_𝑛^𝑘−1=𝑋_𝑚=𝑚^𝜖,(2.5) and so ^𝑚_𝑛=1𝑤^𝛼_𝑛^𝑘𝑛=^𝑚_𝑛=1𝑋_𝑛−𝑋_𝑛−1𝑋_𝑛^𝑘−1=^𝑚_𝑛=1𝑛^𝜖−(𝑛−1)^𝜖𝑛^{𝜖(𝑘−1)}≥𝜖^𝑚_𝑛=1(𝑛−1)^𝜖−1𝑛^{𝜖(𝑘−1)}=𝜖^𝑚_𝑛=1(𝑛−1)^𝜖𝑘−1∼𝑚^𝜖𝑘𝑘as𝑚→∞.(2.6) This is because 𝑣^𝜖−1≥𝑛^𝜖−1 for 𝑛−1≤𝑣≤𝑛.
This shows that, when 𝑘>1, (1.11) implies (2.1) but not conversely.
We need the following lemmas for the proof of our theorem.

Lemma 2.5 (see [7]). If 0<𝛼≤1 and 1≤𝑣≤𝑛, then ^𝑣_𝑝=0𝐴^𝛼−1_𝑛−𝑝𝑎_𝑝≤max_{1≤𝑚≤𝑣}^𝑚_𝑝=0𝐴^𝛼−1_𝑚−𝑝𝑎_𝑝.(2.7)

Lemma 2.6 (see [8]). If −1<𝛼≤𝛽, 𝑘>1 and the series ∑𝑎_𝑛 is summable |𝐶,𝛼|_𝑘, then it is also summable |𝐶,𝛽|_𝑘.

Lemma 2.7 (see [9]). Let (𝑋_𝑛) be an almost increasing sequence such that 𝑛|Δ𝑋_𝑛|=𝑂(𝑋_𝑛).
If (𝐴_𝑛) is a 𝛿-quasimonotone with ∑𝑛𝛿_𝑛𝑋_𝑛<∞, ∑𝐴_𝑛𝑋_𝑛 is convergent, then 𝑛𝐴_𝑛𝑋_𝑛=𝑂(1)𝑎𝑠𝑛→∞,_∞𝑛=1𝑛𝑋_𝑛Δ𝐴_𝑛<∞.(2.8)

Lemma 2.8 (see [10]). Let 𝑝₀>0, 𝑝_𝑛≥0, and (𝑝_𝑛) be a nonincreasing sequence. If the series ∑𝑎_𝑛 is summable |𝐶,1|_𝑘, then the series ∑𝑎_𝑛𝑃_𝑛(𝑛+1)⁻¹ is summable |𝑁,𝑝_𝑛|_𝑘, 𝑘≥1.

3. Proof of Theorem 2.1

Let (𝑇^𝛼_𝑛) be the 𝑛th (𝐶,𝛼), with 0<𝛼≤1, mean of the sequence (𝑛𝑎_𝑛𝜆_𝑛). Then, by (1.2), we have 𝑇^𝛼_𝑛=1𝐴^𝛼_𝑛^𝑛_𝑣=1𝐴^𝛼−1_𝑛−𝑣𝑣𝑎_𝑣𝜆_𝑣.(3.1) First applying Abel's transformation and then using Lemma 2.5, we have that 𝑇^𝛼_𝑛=1𝐴^𝛼_𝑛^𝑛−1_𝑣=1Δ𝜆_𝑣^𝑣_𝑝=1𝐴^𝛼−1_𝑛−𝑝𝑝𝑎_𝑝+𝜆_𝑛𝐴^𝛼_𝑛^𝑛_𝑣=1𝐴^𝛼−1_𝑛−𝑣𝑣𝑎_𝑣,𝑇(3.2)_𝛼𝑛≤1𝐴^𝛼_𝑛^𝑛−1_𝑣=1Δ𝜆_𝑣^𝑣_𝑝=1𝐴^𝛼−1_𝑛−𝑝𝑝𝑎_𝑝+𝜆_𝑛𝐴^𝛼_𝑛^𝑛_𝑣=1𝐴^𝛼−1_𝑛−𝑣𝑣𝑎_𝑣≤1𝐴^𝛼_𝑛^𝑛−1_𝑣=1𝐴^𝛼_𝑣𝑤^𝛼_𝑣Δ𝜆_𝑣+𝜆_𝑛𝑤^𝛼_𝑛=𝑇^𝛼_𝑛,1+𝑇^𝛼_𝑛,2.(3.3) To complete the proof of Theorem 2.1, by Minkowski's inequality, it is sufficient to show that ^∞_𝑛=1𝑛⁻¹𝑇^𝛼_𝑛,𝑟^𝑘<∞for𝑟=1,2.(3.4) Whenever 𝑘>1, we can apply Hölder's inequality with indices 𝑘 and 𝑘^′, where (1/𝑘)+(1/𝑘^′)=1, we get that ^𝑚+1_𝑛=2𝑛⁻¹𝑇^𝛼_𝑛,1^𝑘≤^𝑚+1_𝑛=2𝑛⁻¹𝐴^𝛼_𝑛^−𝑘𝑚_𝑣=1𝐴^𝛼_𝑣𝑤^𝛼_𝑣Δ𝜆_𝑣^𝑘≤^𝑚+1_𝑛=2𝑛^{−1−𝛼𝑘𝑛−1}_𝑣=1𝑣^𝛼𝑘𝑤^𝛼_𝑣^𝑘𝐴_𝑣^𝑘×^𝑛−1_𝑣=11^𝑘−1=𝑂(1)^𝑚_𝑣=1𝑣^𝛼𝑘𝑤^𝛼_𝑣^𝑘𝐴_𝑣𝐴_𝑣^{𝑘−1𝑚+1}_𝑛=𝑣+11𝑛^{2+(𝛼−1)𝑘}=𝑂(1)^𝑚_𝑣=1𝑣^𝛼𝑘𝑤^𝛼_𝑣^𝑘𝐴_𝑣1𝑣𝑋_𝑣^{𝑘−1∞𝑣}𝑑𝑥𝑥^{2+(𝛼−1)𝑘}=𝑂(1)^𝑚_𝑣=1𝑣𝐴_𝑣𝑤^𝛼_𝑣^𝑘𝑣𝑋_𝑣^𝑘−1=𝑂(1)^𝑚−1_𝑣=1Δ𝑣𝐴_𝑣^𝑣_𝑟=1𝑤^𝛼_𝑟^𝑘𝑟𝑋_𝑟^𝑘−1+𝑂(1)𝑚𝐴_𝑚^𝑚_𝑣=1𝑤^𝛼_𝑣^𝑘𝑣𝑋_𝑣^𝑘−1=𝑂(1)^𝑚−1_𝑣=1Δ𝑣𝐴_𝑣𝑋_𝑣+𝑂(1)𝑚𝐴_𝑚𝑋_𝑚=𝑂(1)^𝑚−1_𝑣=1𝑣Δ𝐴_𝑣𝑋_𝑣+𝑂(1)^𝑚_𝑣=1𝐴_𝑣𝑋_𝑣+𝑂(1)𝑚𝐴_𝑚𝑋_𝑚=𝑂(1)as𝑚→∞,(3.5) by virtue of the hypotheses of Theorem 2.1 and Lemma 2.7. Again, we have that ^𝑚_𝑛=1𝑛⁻¹𝑇^𝛼_𝑛,2^𝑘=𝑂(1)^𝑚_𝑛=1𝜆_𝑛𝜆_𝑛^𝑘−1𝑤^𝛼_𝑛^𝑘𝑛=𝑂(1)^𝑚_𝑛=1𝜆_𝑛𝑤^𝛼_𝑛^𝑘𝑛𝑋_𝑛^𝑘−1=𝑂(1)^𝑚−1_𝑛=1Δ𝜆_𝑛^𝑛_𝑣=1𝑤^𝛼_𝑣^𝑘𝑣𝑋_𝑣^𝑘−1+𝑂(1)𝜆_𝑚^𝑚_𝑛=1𝑤^𝛼_𝑛^𝑘𝑛𝑋_𝑛^𝑘−1=𝑂(1)^𝑚−1_𝑛=1Δ𝜆_𝑛𝑋_𝑛+𝑂(1)𝜆_𝑚𝑋_𝑚=𝑂(1)^𝑚−1_𝑛=1𝐴_𝑛𝑋_𝑛+𝑂(1)𝜆_𝑚𝑋_𝑚=𝑂(1)as𝑚→∞,(3.6) by virtue of the hypotheses of Theorem 2.1. This completes the proof of Theorem 2.1. If we take 𝛼=1, then we get a new result dealing with |𝐶,1|_𝑘 summability factors.

Proof of Theorem 2.2. In order to prove Theorem 2.2, we need to consider only the special case in which (𝑁,𝑝_𝑛) is (𝐶,𝛼). Therefore, Theorem 2.2 will then follow by means of Theorem 2.1, Lemma 2.6 (for 𝛽=1), and Lemma 2.8. If we take 𝛼=1, then we get a new result for the absolute Nörlund summability factors of infinite series.

Acknowledgment

The author wishes his sincerest thanks to the referee for invaluable suggestions for the improvement of this paper.