Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 4(2010), Pages 83-86.
ON THE GENERALIZED ABSOLUTE CES `ARO SUMMABILITY
(DEDICATED IN OCCASION OF THE 70-YEARS OF PROFESSOR HARI M. SRIVASTAVA)
H ¨USEYIN BOR, DANSHENG YU
Abstract. In this paper, a known theorem dealing with∣𝐶, 𝛼∣summability factors, has been generalized for∣𝐶, 𝛼, 𝛽∣𝑘 summability factors. Some new results have also been obtained.
1. Introduction
Let∑𝑎𝑛 be a given infinite series with partial sums (𝑠𝑛). We denote by 𝑢𝛼,𝛽𝑛 and𝑡𝛼,𝛽𝑛 the n-th Ces`aro means of order (𝛼, 𝛽), with 𝛼+𝛽 >−1, of the sequence (𝑠𝑛) and (𝑛𝑎𝑛), respectively, i.e., (see [2])
𝑢𝛼,𝛽𝑛 = 1 𝐴𝛼+𝛽𝑛
𝑛
∑
𝑣=0
𝐴𝛼−1𝑛−𝑣𝐴𝛽𝑣𝑠𝑣 (1.1)
𝑡𝛼,𝛽𝑛 = 1 𝐴𝛼+𝛽𝑛
𝑛
∑
𝑣=1
𝐴𝛼−1𝑛−𝑣𝐴𝛽𝑣𝑣𝑎𝑣, (1.2) where
𝐴𝛼+𝛽𝑛 =𝑂(𝑛𝛼+𝛽), 𝐴𝛼+𝛽0 = 1 𝑎𝑛𝑑 𝐴𝛼+𝛽−𝑛 = 0 𝑓 𝑜𝑟 𝑛 >0. (1.3) The series∑𝑎𝑛 is said to be summable∣𝐶, 𝛼, 𝛽∣𝑘, 𝑘≥1 and𝛼+𝛽 >−1, if (see [4])
∞
∑
𝑛=1
𝑛𝑘−1∣𝑢𝛼,𝛽𝑛 −𝑢𝛼,𝛽𝑛−1∣<∞. (1.4) Since𝑡𝛼,𝛽𝑛 =𝑛(𝑢𝛼,𝛽𝑛 −𝑢𝛼,𝛽𝑛−1) (see [4]), condition (4) can also be written as
∞
∑
𝑛=1
1
𝑛 ∣𝑡𝛼,𝛽𝑛 ∣𝑘<∞. (1.5) If we take𝛽 = 0, then∣𝐶, 𝛼, 𝛽∣𝑘 summability reduces to∣𝐶, 𝛼∣𝑘 summability (see [5]). It should be noted that obviously (𝐶, 𝛼,0) mean is the same as (𝐶, 𝛼)
2000Mathematics Subject Classification. 40D15, 40F05, 40G05, 40G99.
Key words and phrases. Absolute Ces`aro summability, infinite series, summability factors.
c
⃝2010 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted September 2, 2010. Published October 23, 2010.
83
84 H ¨USEYIN BOR, DANSHENG YU
mean. A sequence (𝜆𝑛) is said to be convex if Δ2𝜆𝑛 ≥0, where Δ2𝜆𝑛 = Δ𝜆𝑛− Δ𝜆𝑛+1.
Pati [6] has proved the following theorem dealing with∣𝐶, 𝛼∣ summability fac- tors.
Theorem 1.1. If (𝜆𝑛) is a convex sequence such that∑
𝑛−1𝜆𝑛 is convergent and the sequence (𝜃𝑛𝛼)defined by
𝜃𝛼𝑛 =∣𝑡𝛼𝑛 ∣, 𝛼= 1, (1.6)
𝜃𝛼𝑛 = max
1≤𝑣≤𝑛∣𝑡𝛼𝑣 ∣, 0< 𝛼 <1 (1.7) satisfies the condition
𝜃𝑛𝛼=𝑂(1)(𝐶,1), (1.8)
then the series∑
𝑎𝑛𝜆𝑛 is summable∣𝐶, 𝛼∣for0< 𝛼≤1.
2. The Main Result
The aim of this paper is to generalize Theorem 1.1 for∣𝐶, 𝛼, 𝛽 ∣𝑘 summability.
We shall prove the following theorem.
Theorem 2.1. If (𝜆𝑛) is a convex sequence such that∑𝑛−1𝜆𝑛 is convergent and the sequence (𝜃𝑛𝛼,𝛽)defined by
𝜃𝛼,𝛽𝑛 =∣𝑡𝛼,𝛽𝑛 ∣, 𝛼= 1, 𝛽 >−1 (2.1) 𝜃𝛼,𝛽𝑛 = max
1≤𝑣≤𝑛∣𝑡𝛼,𝛽𝑣 ∣, 0< 𝛼 <1, 𝛽 >−1 (2.2) satisfies the condition
(𝜃𝑛𝛼,𝛽)𝑘=𝑂(1)(𝐶,1), (2.3) then the series∑𝑎𝑛𝜆𝑛 is summable∣𝐶, 𝛼, 𝛽∣𝑘 for0< 𝛼≤1,𝛽 >−1 and𝑘≥1.
It should be noted that if we take𝑘= 1 and 𝛽= 0, then we get Theorem 1.1.
We need the following lemmas for the proof of our theorem.
Lemma 2.2. ([3]) If (𝜆𝑛) is a convex sequence such that ∑
𝑛−1𝜆𝑛 is convergent, then
𝑛Δ𝜆𝑛 →0,
∞
∑
𝑛=1
(𝑛+ 1)Δ2𝜆𝑛
is convergent.
Lemma 2.3. ([1]). If0< 𝛼≤1,𝛽 >−1 and1≤𝑣≤𝑛, then
∣
𝑣
∑
𝑝=0
𝐴𝛼−1𝑛−𝑝𝐴𝛽𝑝𝑎𝑝∣≤ max
1≤𝑚≤𝑣∣
𝑚
∑
𝑝=0
𝐴𝛼−1𝑚−𝑝𝐴𝛽𝑝𝑎𝑝∣. (2.4)
Proof of the theorem. Let (𝑇𝑛𝛼,𝛽) be the n-th (𝐶, 𝛼, 𝛽) mean of the sequence (𝑛𝑎𝑛𝜆𝑛). Then, by (1.2), we have
𝑇𝑛𝛼,𝛽 = 1 𝐴𝛼+𝛽𝑛
𝑛
∑
𝑣=1
𝐴𝛼−1𝑛−𝑣𝐴𝛽𝑣𝑣𝑎𝑣𝜆𝑣.
ON THE GENERALIZED ABSOLUTE CES `ARO SUMMABILITY 85
First, applying Abel’s transformation and then using Lemma 2.3, we have that 𝑇𝑛𝛼,𝛽 = 1
𝐴𝛼+𝛽𝑛 𝑛−1
∑
𝑣=1
Δ𝜆𝑣 𝑣
∑
𝑝=1
𝐴𝛼−1𝑛−𝑝𝐴𝛽𝑝𝑝𝑎𝑝+ 𝜆𝑛
𝐴𝛼+𝛽𝑛 𝑛
∑
𝑣=1
𝐴𝛼−1𝑛−𝑣𝐴𝛽𝑣𝑣𝑎𝑣,
thus,
∣𝑇𝑛𝛼,𝛽∣ ≤ 1 𝐴𝛼+𝛽𝑛
𝑛−1
∑
𝑣=1
∣Δ𝜆𝑣 ∣∣
𝑣
∑
𝑝=1
𝐴𝛼−1𝑛−𝑝𝐴𝛽𝑝𝑝𝑎𝑝∣+∣𝜆𝑛∣ 𝐴𝛼+𝛽𝑛
∣
𝑛
∑
𝑣=1
𝐴𝛼−1𝑛−𝑣𝐴𝛽𝑣𝑣𝑎𝑣∣
≤ 1
𝐴𝛼+𝛽𝑛 𝑛−1
∑
𝑣=1
𝐴𝛼𝑣𝐴𝛽𝑣𝜃𝑣𝛼,𝛽∣Δ𝜆𝑣∣+∣𝜆𝑛∣𝜃𝛼,𝛽𝑛
= 𝑇𝑛,1𝛼,𝛽+𝑇𝑛,2𝛼,𝛽, 𝑠𝑎𝑦.
Since
∣𝑇𝑛,1𝛼,𝛽+𝑇𝑛,2𝛼,𝛽 ∣𝑘≤2𝑘(∣𝑇𝑛,1𝛼,𝛽∣𝑘 +∣𝑇𝑛,2𝛼,𝛽∣𝑘),
in order to complete the proof of the theorem, by (5), it is sufficient to show that
∞
∑
𝑛=1
1
𝑛 ∣𝑇𝑛,𝑟𝛼,𝛽∣𝑘<∞ 𝑓 𝑜𝑟 𝑟= 1,2.
Whenever 𝑘 > 1, we can apply H¨older’s inequality with indices 𝑘 and 𝑘′, where
1
𝑘 +𝑘1′ = 1, we get that
𝑚+1
∑
𝑛=2
1 𝑛 𝑇𝑛,1𝛼,𝛽
𝑘
≤
𝑚+1
∑
𝑛=2
1 𝑛
1 𝐴𝛼+𝛽𝑛
𝑛−1
∑
𝑣=1
𝐴𝛼𝑣𝐴𝛽𝑣𝜃𝛼,𝛽𝑣 Δ𝜆𝑣
𝑘
= 𝑂(1)
𝑚+1
∑
𝑛=2
1 𝑛1+(𝛼+𝛽)𝑘
{𝑛−1
∑
𝑣=1
𝑣𝛼𝑘𝑣𝛽𝑘Δ𝜆𝑣(𝜃𝛼,𝛽𝑣 )𝑘 }
× {𝑛−1
∑
𝑣=1
Δ𝜆𝑣
}𝑘−1
= 𝑂(1)
𝑚
∑
𝑣=1
𝑣(𝛼+𝛽)𝑘Δ𝜆𝑣(𝜃𝛼,𝛽𝑣 )𝑘
𝑚+1
∑
𝑛=𝑣+1
1 𝑛1+(𝛼+𝛽)𝑘
= 𝑂(1)
𝑚
∑
𝑣=1
𝑣(𝛼+𝛽)𝑘Δ𝜆𝑣(𝜃𝛼,𝛽𝑣 )𝑘
∫ ∞
𝑣
𝑑𝑥 𝑥1+(𝛼+𝛽)𝑘
= 𝑂(1)
𝑚
∑
𝑣=1
Δ𝜆𝑣(𝜃𝛼,𝛽𝑣 )𝑘
= 𝑂(1)
𝑚−1
∑
𝑣=1
Δ(Δ𝜆𝑣)
𝑣
∑
𝑝=1
(𝜃𝑝𝛼,𝛽)𝑘+𝑂(1)Δ𝜆𝑚 𝑚
∑
𝑣=1
(𝜃𝛼,𝛽𝑣 )𝑘
= 𝑂(1)
𝑚
∑
𝑣=1
𝑣Δ2𝜆𝑣+𝑂(1)𝑚Δ𝜆𝑚
= 𝑂(1) 𝑎𝑠 𝑚→ ∞,
in view of hypotheses of the theorem and Lemma 2.2.
86 H ¨USEYIN BOR, DANSHENG YU
Similarly , we have that
𝑚
∑
𝑛=1
1
𝑛 ∣𝜆𝑛𝜃𝛼,𝛽𝑛 ∣𝑘 = 𝑂(1)
𝑚
∑
𝑛=1
𝜆𝑛
𝑛 (𝜃𝑛𝛼,𝛽)𝑘
= 𝑂(1)
𝑚−1
∑
𝑛=1
Δ(𝑛−1𝜆𝑛)
𝑛
∑
𝑣=1
(𝜃𝑣𝛼,𝛽)𝑘
+ 𝑂(1)𝜆𝑚 𝑚
𝑚
∑
𝑣=1
(𝜃𝑣𝛼,𝛽)𝑘
= 𝑂(1)
𝑚−1
∑
𝑛=1
Δ𝜆𝑛+𝑂(1)
𝑚−1
∑
𝑛=1
𝜆𝑛+1
𝑛+ 1+𝑂(1)𝜆𝑚
= 𝑂(1)(𝜆1−𝜆𝑚) +𝑂(1)
𝑚−1
∑
𝑛=1
𝜆𝑛+1
𝑛+ 1 +𝑂(1)𝜆𝑚
= 𝑂(1) 𝑎𝑠 𝑚→ ∞.
Therefore, by (1.5), we get that
∞
∑
𝑛=1
1
𝑛 ∣𝑇𝑛,𝑟𝛼,𝛽∣𝑘<∞ 𝑓 𝑜𝑟 𝑟= 1,2.
This completes the proof of the theorem. If we take𝛽 = 0, then we get a new result for ∣𝐶, 𝛼∣𝑘 summability factors. Also, if we take 𝛽 = 0 and 𝛼= 1 , then we get another new result for∣𝐶,1∣𝑘 summability factors.
References
[1] H. Bor, On a new application of quasi power increasing sequences,Proc. Estonian Acad. Sci.
Phys. Math.57, (2008), 205-209.
[2] D. Borwein, Theorems on some methods of summability, Quart. J. Math., Oxford,Ser.9, (1958), 310-316.
[3] H. C. Chow , On the summability factors of Fourier Series,J. London Math. Soc.16, (1941), 215-220.
[4] G. Das, A Tauberian theorem for absolute summability,Proc. Camb. Phil. Soc.67, (1970), 321-326.
[5] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley,Proc. London Math. Soc.7, (1957), 113-141.
[6] T. Pati, The summability factors of infinite series,Duke Math. J.21, (1954), 271-284.
H¨useyin Bor
Department of Mathematics, Bahc¸elievler, P.O.Box 121 , 06502, Ankara, Turkey
E-mail address:[email protected]
Dansheng Yu
Department of Mathematics, Hangzhou Normal University, Hangzhou Zhejiang 310036 , China
E-mail address:[email protected]
