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 CESÀRO SUMMABILITY
(DEDICATED IN OCCASION OF THE 70-YEARS OF
PROFESSOR HARI M. SRIVASTAVA)
HÜSEYIN 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,𝛽
𝑛
𝛼,𝛽
and 𝑡𝑛 the n-th Cesàro means of order (𝛼, 𝛽), with 𝛼 + 𝛽 > −1, of the sequence
(𝑠𝑛 ) and (𝑛𝑎𝑛 ), respectively, i.e., (see [2])
∑
𝑢𝛼,𝛽
=
𝑛
𝑡𝛼,𝛽
𝑛
=
1
𝑛
∑
𝐴𝑛𝛼+𝛽
𝑣=0
1
𝑛
∑
𝐴𝑛𝛼+𝛽 𝑣=1
𝛼−1 𝛽
𝐴𝑣 𝑠𝑣
𝐴𝑛−𝑣
(1.1)
𝛼−1 𝛽
𝐴𝑛−𝑣
𝐴𝑣 𝑣𝑎𝑣 ,
(1.2)
where
𝛼+𝛽
𝐴𝛼+𝛽
= 𝑂(𝑛𝛼+𝛽 ), 𝐴𝛼+𝛽
= 1 𝑎𝑛𝑑 𝐴−𝑛
= 0 𝑓 𝑜𝑟 𝑛 > 0.
(1.3)
𝑛
0
∑
The series
𝑎𝑛 is said to be summable ∣ 𝐶, 𝛼, 𝛽 ∣𝑘 , 𝑘 ≥ 1 and 𝛼 + 𝛽 > −1, if (see
[4])
∞
∑
𝛼,𝛽
𝑛𝑘−1 ∣ 𝑢𝛼,𝛽
(1.4)
𝑛 − 𝑢𝑛−1 ∣< ∞.
𝑛=1
𝛼,𝛽
Since 𝑡𝛼,𝛽
= 𝑛(𝑢𝛼,𝛽
𝑛
𝑛 − 𝑢𝑛−1 ) (see [4]), condition (4) can also be written as
∞
∑
1 𝛼,𝛽 𝑘
∣ 𝑡𝑛 ∣ < ∞.
𝑛
𝑛=1
(1.5)
If we take 𝛽 = 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)
2000 Mathematics Subject Classification. 40D15, 40F05, 40G05, 40G99.
Key words and phrases. Absolute Cesàro summability, infinite series, summability factors.
c
⃝2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted September 2, 2010. Published October 23, 2010.
83
84
HÜSEYIN BOR, DANSHENG YU
mean. A sequence (𝜆𝑛 ) is said to be convex if Δ2 𝜆𝑛 ≥ 0, where Δ2 𝜆𝑛 = Δ𝜆𝑛 −
Δ𝜆𝑛+1 .
Pati [6] has proved the following theorem dealing with ∣ 𝐶, 𝛼 ∣ summability factors.
∑ −1
Theorem 1.1. If (𝜆𝑛 ) is a convex sequence such that
𝑛 𝜆𝑛 is convergent and
the sequence (𝜃𝑛𝛼 ) defined by
𝜃𝑛𝛼 =∣ 𝑡𝛼
𝑛 ∣, 𝛼 = 1,
(1.6)
𝜃𝑛𝛼 = max ∣ 𝑡𝛼
𝑣 ∣, 0 < 𝛼 < 1
(1.7)
𝜃𝑛𝛼 = 𝑂(1)(𝐶, 1),
(1.8)
1≤𝑣≤𝑛
satisfies the condition
then the series
∑
𝑎𝑛 𝜆𝑛 is summable ∣ 𝐶, 𝛼 ∣ for 0 < 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.
∑ −1
Theorem 2.1. If (𝜆𝑛 ) is a convex sequence such that
𝑛 𝜆𝑛 is convergent and
the sequence (𝜃𝑛𝛼,𝛽 ) defined by
𝜃𝑛𝛼,𝛽 =∣ 𝑡𝛼,𝛽
∣, 𝛼 = 1, 𝛽 > −1
𝑛
(2.1)
𝜃𝑛𝛼,𝛽 = max ∣ 𝑡𝛼,𝛽
∣, 0 < 𝛼 < 1, 𝛽 > −1
𝑣
(2.2)
1≤𝑣≤𝑛
satisfies the condition
(𝜃𝑛𝛼,𝛽 )𝑘 = 𝑂(1)(𝐶, 1),
(2.3)
∑
then the series
𝑎𝑛 𝜆𝑛 is summable ∣ 𝐶, 𝛼, 𝛽 ∣𝑘 for 0 < 𝛼 ≤ 1, 𝛽 > −1 and 𝑘 ≥ 1.
It should be noted that if we take 𝑘 = 1 and 𝛽 = 0, then we get Theorem 1.1.
We need the following lemmas for the proof of our theorem.
∑ −1
Lemma 2.2. ([3]) If (𝜆𝑛 ) is a convex sequence such that
𝑛 𝜆𝑛 is convergent,
then
𝑛Δ𝜆𝑛 → 0,
∞
∑
(𝑛 + 1)Δ2 𝜆𝑛
𝑛=1
is convergent.
Lemma 2.3. ([1]). If 0 < 𝛼 ≤ 1, 𝛽 > −1 and 1 ≤ 𝑣 ≤ 𝑛, then
∣
𝑣
∑
𝛽
𝐴𝛼−1
𝑛−𝑝 𝐴𝑝 𝑎𝑝 ∣≤ max ∣
1≤𝑚≤𝑣
𝑝=0
𝑚
∑
𝛼−1 𝛽
𝐴𝑚−𝑝
𝐴𝑝 𝑎𝑝 ∣ .
(2.4)
𝑝=0
Proof of the theorem. Let (𝑇𝑛𝛼,𝛽 ) be the n-th (𝐶, 𝛼, 𝛽) mean of the sequence
(𝑛𝑎𝑛 𝜆𝑛 ). Then, by (1.2), we have
𝑇𝑛𝛼,𝛽
=
1
𝑛
∑
𝐴𝑛𝛼+𝛽
𝑣=1
𝛼−1 𝛽
𝐴𝑛−𝑣
𝐴𝑣 𝑣𝑎𝑣 𝜆𝑣 .
ON THE GENERALIZED ABSOLUTE CESÀRO SUMMABILITY
85
First, applying Abel’s transformation and then using Lemma 2.3, we have that
𝑇𝑛𝛼,𝛽
=
1
𝑛−1
∑
𝐴𝛼+𝛽
𝑛
𝑣=1
Δ𝜆𝑣
𝑣
∑
𝛼−1 𝛽
𝐴𝑛−𝑝
𝐴𝑝 𝑝𝑎𝑝
𝑝=1
+
𝑛
𝜆𝑛 ∑
𝐴𝑛𝛼+𝛽
𝛼−1 𝛽
𝐴𝑛−𝑣
𝐴𝑣 𝑣𝑎𝑣 ,
𝑣=1
thus,
∣ 𝑇𝑛𝛼,𝛽 ∣
≤
≤
=
𝑛−1
∑
1
𝐴𝛼+𝛽
𝑛
𝑣
∑
∣ Δ𝜆𝑣 ∣∣
𝑣=1
𝛽
𝐴𝛼−1
𝑛−𝑝 𝐴𝑝 𝑝𝑎𝑝 ∣ +
𝑝=1
𝑛−1
∑
1
𝛽 𝛼,𝛽
𝐴𝛼
𝑣 𝐴𝑣 𝜃𝑣
𝛼+𝛽
𝐴𝑛
𝑣=1
𝛼,𝛽
𝛼,𝛽
𝑇𝑛,1
+ 𝑇𝑛,2
, 𝑠𝑎𝑦.
∣ 𝜆𝑛 ∣
𝐴𝑛𝛼+𝛽
∣
𝑛
∑
𝛼−1 𝛽
𝐴𝑛−𝑣
𝐴𝑣 𝑣𝑎𝑣 ∣
𝑣=1
∣ Δ𝜆𝑣 ∣ + ∣ 𝜆𝑛 ∣ 𝜃𝑛𝛼,𝛽
Since
𝛼,𝛽
𝛼,𝛽 𝑘
𝛼,𝛽 𝑘
𝛼,𝛽 𝑘
∣ 𝑇𝑛,1
+ 𝑇𝑛,2
∣ ≤ 2𝑘 (∣ 𝑇𝑛,1
∣ + ∣ 𝑇𝑛,2
∣ ),
in order to complete the proof of the theorem, by (5), it is sufficient to show that
∞
∑
1
𝛼,𝛽 𝑘
∣ 𝑇𝑛,𝑟
∣ < ∞ 𝑓 𝑜𝑟
𝑛
𝑛=1
𝑟 = 1, 2.
Whenever 𝑘 > 1, we can apply Hölder’s inequality with indices 𝑘 and 𝑘 ′ , where
1
1
𝑘 + 𝑘′ = 1, we get that
𝑘
𝑚+1
𝑚+1
∑
∑ 1 𝛼,𝛽 𝑘
∑ 1 1 𝑛−1
𝛼 𝛽 𝛼,𝛽
𝐴𝑣 𝐴𝑣 𝜃𝑣 Δ𝜆𝑣 ≤
𝑇𝑛,1 𝛼+𝛽
𝐴𝑛
𝑛
𝑛
𝑣=1
𝑛=2
𝑛=2
{
}
𝑚+1
𝑛−1
∑
∑
1
= 𝑂(1)
𝑣 𝛼𝑘 𝑣 𝛽𝑘 Δ𝜆𝑣 (𝜃𝑣𝛼,𝛽 )𝑘
1+(𝛼+𝛽)𝑘
𝑛
𝑛=2
𝑣=1
{𝑛−1
}𝑘−1
∑
×
Δ𝜆𝑣
𝑣=1
=
=
=
𝑚
∑
𝑚+1
∑
1
1+(𝛼+𝛽)𝑘
𝑛
𝑣=1
𝑛=𝑣+1
∫ ∞
𝑚
∑
𝑑𝑥
𝑂(1)
𝑣 (𝛼+𝛽)𝑘 Δ𝜆𝑣 (𝜃𝑣𝛼,𝛽 )𝑘
1+(𝛼+𝛽)𝑘
𝑥
𝑣
𝑣=1
𝑂(1)
𝑂(1)
𝑚
∑
𝑣 (𝛼+𝛽)𝑘 Δ𝜆𝑣 (𝜃𝑣𝛼,𝛽 )𝑘
Δ𝜆𝑣 (𝜃𝑣𝛼,𝛽 )𝑘
𝑣=1
=
𝑂(1)
𝑚−1
∑
𝑣=1
=
𝑂(1)
=
𝑂(1)
𝑚
∑
Δ(Δ𝜆𝑣 )
𝑣
𝑚
∑
∑
(𝜃𝑝𝛼,𝛽 )𝑘 + 𝑂(1)Δ𝜆𝑚
(𝜃𝑣𝛼,𝛽 )𝑘
𝑝=1
𝑣Δ2 𝜆𝑣 + 𝑂(1)𝑚Δ𝜆𝑚
𝑣=1
𝑎𝑠
𝑚 → ∞,
in view of hypotheses of the theorem and Lemma 2.2.
𝑣=1
86
HÜSEYIN BOR, DANSHENG YU
Similarly , we have that
𝑚
∑
1
∣ 𝜆𝑛 𝜃𝑛𝛼,𝛽 ∣𝑘 =
𝑛
𝑛=1
=
𝑂(1)
𝑂(1)
𝑚
∑
𝜆𝑛 𝛼,𝛽 𝑘
(𝜃 )
𝑛 𝑛
𝑛=1
𝑚−1
∑
−1
Δ(𝑛
𝑛
∑
𝜆𝑛 )
(𝜃𝑣𝛼,𝛽 )𝑘
𝑛=1
+ 𝑂(1)
=
𝑂(1)
𝜆𝑚
𝑚
𝑣=1
𝑚
∑
(𝜃𝑣𝛼,𝛽 )𝑘
𝑣=1
𝑚−1
∑
Δ𝜆𝑛 + 𝑂(1)
𝑛=1
𝑚−1
∑
𝑛=1
=
𝑂(1)(𝜆1 − 𝜆𝑚 ) + 𝑂(1)
=
𝑂(1) 𝑎𝑠
𝜆𝑛+1
+ 𝑂(1)𝜆𝑚
𝑛+1
𝑚−1
∑
𝑛=1
𝜆𝑛+1
+ 𝑂(1)𝜆𝑚
𝑛+1
𝑚 → ∞.
Therefore, by (1.5), we get that
∞
∑
1
𝛼,𝛽 𝑘
∣ 𝑇𝑛,𝑟
∣ < ∞ 𝑓 𝑜𝑟
𝑛
𝑛=1
𝑟 = 1, 2.
This completes the proof of the theorem. If we take 𝛽 = 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.
Hüseyin Bor
Department of Mathematics, Bahçelievler, P.O.Box 121 , 06502,
Ankara, Turkey
E-mail address: hbor33@gmail.com
Dansheng Yu
Department of Mathematics, Hangzhou Normal University,
Hangzhou Zhejiang 310036 , China
E-mail address: anshengyu@yahoo.com.cn