WOS: 000296987000016Let (s(n)) be a sequence of real numbers such that lim sup(n) sigma(n) = beta and lim inf(n) sigma(n) = alpha, where sigma(n) = 1/n Sigma(n)(k=1) s(k) and beta not equal alpha. We prove that lim sup(n) s(n) = beta and lim inf(n) s(n) = alpha if the following conditions hold: lim(n) inf 1/[lambda n] - n Sigma([lambda n])(k=n+1) (s(k) - s(n)) >= (beta - alpha) lambda/lambda - 1 for lambda > 1, lim(n) inf 1/n - [lambda n] Sigma(n)(k=[lambda n]+1) (s(k) - s(k)) >= (beta - alpha) lambda/1 -lambda for 0 < lambda < 1, where [lambda n] denotes the integer part of lambda n. (C) 2011 Elsevier Ltd. All right reserved

Canak, Ibrahim

Computers & Mathematics with Applications

AbstractLet (sn) be a sequence of real numbers such that lim supnσn=β and lim infnσn=α, where σn=1n∑k=1nsk and β≠α. We prove that lim supnsn=β and lim infnsn=α if the following conditions hold: lim infn1[λn]−n∑k=n+1[λn](sk−sn)≥(β−α)λλ−1forλ>1,lim infn1n−[λn]∑k=[λn]+1n(sn−sk)≥(β−α)λ1−λfor0<λ<1, where [λn] denotes the integer part of λn

Çanak, İbrahi̇m

Elsevier - Publisher Connector 

Computers and Mathematics with Applications 62 (2011) 3446–3448
Contents lists available at SciVerse ScienceDirect
Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
On (C, 1)means of sequences
İbrahim Çanak ∗
Department of Mathematics, Ege University, 35100 İzmir, Turkey
a r t i c l e i n f o
Article history:
Received 10 June 2011
Accepted 22 August 2011
Keywords:
(C, 1) bounded sequences
(C, 1) summable sequences
a b s t r a c t
Let (sn) be a sequence of real numbers such that lim supn σn = β and lim infn σn = α,
where σn = 1n
∑n
k=1 sk and β ≠ α. We prove that lim supn sn = β and lim infn sn = α if
the following conditions hold:
lim inf
n
1
[λn] − n
[λn]−
k=n+1
(sk − sn) ≥ (β − α) λ
λ− 1 for λ > 1,
lim inf
n
1
n− [λn]
n−
k=[λn]+1
(sn − sk) ≥ (β − α) λ1− λ for 0 < λ < 1,
where [λn] denotes the integer part of λn.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Let (sn) be a sequence of real numbers, and for each n define the (C, 1)means by
σn = 1n
n−
k=1
sk (n = 1, 2, . . .). (1.1)
A sequence (sn) is said to be (C, 1) summable to s if lim σn = s. A sequence (sn) is said to be (C, 1) bounded if (σn) is
bounded. It is well-known [1] that
lim inf
n
sn ≤ lim inf
n
σn ≤ lim sup
n
σn ≤ lim sup
n
sn (1.2)
for any sequence (sn) of real numbers. If (sn) is a bounded sequence, then lim infn sn and lim supn sn exist and are finite.
If (sn) is a bounded sequence, (sn) is (C, 1) bounded. So lim infn σn and lim supn σn always exist and are finite. That the
converse of this implication is not generally true follows from the example of the sequence defined by
sn =

k, if n = 2k, k = 1, 2, . . .
−k, if n = 2k− 1, k = 1, 2, . . . .
It is clear that lim infn σn = − 12 and lim supn σn = 0, but lim infn sn = −∞ and lim supn sn = ∞. The main goal of this
paper is to show that if (sn) is a sequence of real numbers such that lim supn σn = β and lim infn σn = α, where β ≠ α, then
∗ Tel.: +90 232 311 1010; fax: +90 232 342 6951.
E-mail addresses: ibrahim.canak@ege.edu.tr, ibrahimcanak@yahoo.com.
0898-1221/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.camwa.2011.08.060
İ Çanak / Computers and Mathematics with Applications 62 (2011) 3446–3448 3447
lim supn sn = β and lim infn sn = α, provided that
lim inf
n
1
[λn] − n
[λn]−
k=n+1
(sk − sn) ≥ (β − α) λ
λ− 1 for λ > 1,
lim inf
n
1
n− [λn]
n−
k=[λn]+1
(sn − sk) ≥ (β − α) λ1− λ for 0 < λ < 1,
where [λn] denotes the integer part of λn.
2. The main result
Theorem 2.1. For a sequence (sn) of real numbers, let lim sup σn = β and lim inf σn = α, where β ≠ α. If
lim inf
n
1
[λn] − n
[λn]−
k=n+1
(sk − sn) ≥ (β − α) λ
λ− 1 for λ > 1, (2.1)
and
lim inf
n
1
n− [λn]
n−
k=[λn]+1
(sn − sk) ≥ (β − α) λ1− λ for 0 < λ < 1, (2.2)
where [λn] denotes the integer part of λn, then lim supn sn = β and lim infn sn = α.
We note that since lim supn σn and lim infn σn exist for a (C, 1) bounded sequence, then lim supn sn and lim infn sn exist
and are equal to lim supn σn and lim infn σn, respectively, provided that (2.1) and (2.2) hold.
Corollary 2.2. Let (sn) be (C, 1) summable to s. If
lim inf
n
1
[λn] − n
[λn]−
k=n+1
(sk − sn) ≥ 0 for λ > 1, (2.3)
and
lim inf
n
1
n− [λn]
n−
k=[λn]+1
(sn − sk) ≥ 0 for 0 < λ < 1, (2.4)
where [λn] denotes the integer part of λn, then (sn) is convergent to s.
The conditions (2.3) and (2.4) can be replaced by the weaker conditions
lim sup
λ→1+
lim inf
n
1
[λn] − n
[λn]−
k=n+1
(sk − sn) ≥ 0 (2.5)
and
lim sup
λ→1−
lim inf
n
1
n− [λn]
n−
k=[λn]+1
(sn − sk) ≥ 0 (2.6)
since they are satisfied for all λ > 1 and all 0 < λ < 1, respectively.
It is shown by Móricz [2] that the conditions (2.5) and (2.6) are Tauberian conditions for (C, 1) summability.
3. A lemma
For the proof of Theorem 2.1 we need the following lemma.
Lemma 3.1 ([3,4]). Let (sn) be a sequence of real numbers.
(i) For λ > 1 and sufficiently large n,
sn − σn = [λn] + 1[λn] − n

σ[λn] − σn
− 1[λn] − n
[λn]−
k=n+1
(sk − sn), (3.1)
where [λn] denotes the integer part of λn.
3448 İ Çanak / Computers and Mathematics with Applications 62 (2011) 3446–3448
(ii) For 0 < λ < 1 and sufficiently large n,
sn − σn = [λn] + 1n− [λn]

σn − σ[λn]
+ 1
n− [λn]
n−
k=[λn]+1
(sn − sk), (3.2)
where [λn] denotes the integer part of λn.
4. Proof of Theorem 2.1
Assume that lim supn σn = β and lim infn σn = α, where β ≠ α. Since
lim
n
[λn] + 1
[λn] − n =
λ
λ− 1 for λ > 1, (4.1)
we obtain
lim sup
n
[λn] + 1
[λn] − n

σ[λn] − σn
 ≤ (β − α) λ
λ− 1 for λ > 1. (4.2)
Taking the lim sup of both sides of (3.1) and using (4.2), we have
lim sup
n
sn ≤ β + (β − α) λ
λ− 1 − lim infn
1
[λn] − n
[λn]−
k=n+1
(sk − sn). (4.3)
Taking (2.1) into account, we obtain
lim sup
n
sn ≤ β. (4.4)
By (1.2),
β ≤ lim sup
n
sn. (4.5)
Combining (4.4) and (4.5), we have lim supn sn = β .
Since
lim
n
[λn] + 1
n− [λn] =
λ
1− λ for 0 < λ < 1, (4.6)
we obtain
lim inf
n
[λn] + 1
n− [λn]

σn − σ[λn]
 ≥ (α − β) λ
1− λ for 0 < λ < 1. (4.7)
Taking the lim inf of both sides of (3.2) and using (4.7), we have
lim inf
n
sn ≥ α + (α − β) λ1− λ + lim infn
1
[λn] − n
[λn]−
k=n+1
(sk − sn). (4.8)
Taking (2.2) into account, we obtain
lim inf
n
sn ≥ α. (4.9)
By (1.2),
lim inf
n
sn ≤ α. (4.10)
Combining (4.9) and (4.10), we have lim infn sn = α. This completes the proof of Theorem 2.1.
References
[1] K. Knopp, Theory and Application of Infinite Series, Blackie & Son, Ltd., Glasgow, 1928.
[2] F.Móricz, Necessary and sufficient Tauberian conditions, underwhich convergence follows from summability (C, 1), Bull. Lond.Math. Soc. 26 (3) (1994)
288–294.
[3] F. Dik, Tauberian theorems for convergence and subsequential convergence with moderately oscillatory behavior, Math. Morav. 5 (2001) 19–56.
[4] M. Dik, Tauberian theorems for sequences with moderately oscillatory control moduli, Math. Morav. 5 (2001) 57–94.


On (C,1) means of sequences 

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