Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 17 (2001), 3–7 www.emis.de/journals ON THE GENERALIZED CES `ARO SUMMABILITY FACTORS

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Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 17 (2001), 3–7

www.emis.de/journals

ON THE GENERALIZED CES `ARO SUMMABILITY FACTORS

H. S. ¨OZARSLAN

Abstract. In this paper a general theorem concerning the ψ− |C, α;δ|_k summability factors of infinite series has been proved.

1. Introduction. A sequence (wn) of positive numbers is said to be δ-quasi monotone, ifwn→0,wn >0 ultimately and ∆wn ≥ −δn, where (δn) is a sequence of positive numbers (see[1]). LetP

an be a given infinite series with partial sums (sn). We defineA^α_n by identity

(1)

∞

X

n=0

A^α_nxⁿ = (1−x)^−α−1. The sequence-to-sequence transformations given by

(2) u^α_n = 1

A^α_n

n

X

v=0

A^α−1_n−vsv

(3) t^α_n = 1

A^α_n

n

X

v=1

A^α−1_n−vvav,

define the (C, α) means of the sequences (sn) and (nan), respectively.

The seriesPan is said to be summable |C, α|_k,k≥1 andα >−1, if (see [3])

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∞

X

n=1

n^k−1|u^α_n−u^α_n−1|^k<∞.

If we take α= 1, then|C, α|_k summability is the same as|C,1|_k summability.

Let (ψ_n) be a sequence of positive real numbers. We say that the series Pa_n is said to be summable ψ− |C, α;δ|_k,k≥1,α >−1 andδ≥0, if

(5)

∞

X

n=1

ψ_n^δk+k−1|u^α_n−u^α_n−1|^k<∞.

But since t^α_n =n(u^α_n−u^α_n−1) (see [4]) condition (5) can also be written as (6)

∞

X

n=1

ψ_n^δk+k−1n^−k |t^α_n |^k<∞.

If we takeδ= 0 andψ_n =n(resp. δ= 0,α= 1 andψ_n=n), thenψ− |C, α;δ|_k summability is the same as |C, α|_k (resp. |C,1|_k) summability.

Remark. Since (ψ_n) is a sequence of positive real numbers the summability

2000Mathematics Subject Classification. 40D15, 40F05, 40G05.

Key words and phrases. Absolute summability factors, infinite series.

3

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methodψ− |C, α;δ|_k is a new method and general than the|C, α;δ|_k summability method. On the other hand|C, α;δ|_k andψ− |C, α;δ|_ksummability methods are different from each other. That is they have got different summability fields.

Therefore, we take the sequence (ψn) instead of n.

2. The following theorem is known.

Theorem A([2]). Let t^α_n be the n-th Ces`aro mean of order α, with α ≥ 1, of the sequence (nan) such that an ≥ 0 for all n ≥ 1 whenever α > 1 and let λn →0 as n→ ∞. Suppose that there exists a sequence of numbers (Bn) such that it is δ-quasi monotone withP

n^αδnlogn <∞, P

Bnlognis convergent and

|∆λn|≤|Bn|for all n.

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m

X

n=1

|∆(n^α)||Bn+1|logn=O(1),

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m

X

n=1

1

n |t^α_n|^k=O(logm) asm→ ∞, then the seriesP

anλn is summable|C, α|_k,k≥1.

3. The aim of this paper is to generalize Theorem A in the following form.

Theorem. Letk≥1 andδ≥0. Lett^α_n be the n-th Ces`aro mean of orderα, with α≥1, of the sequence (nan) such that an ≥0 for all n≥1 wheneverα >1 and let λn → 0 as n→ ∞. Suppose that there exists a sequence of numbers (Bn) such that it isδ-quasi monotone withPn^αδ_nlogn <∞,PB_nlognis convergent,

|∆λ_n|≤|B_n| for all n and that condition (7) of Theorem A is satisfied. If there exists an >0 such that the sequence (n^−kψ^δk+k−1_n ) is non-increasing and (9)

m

X

n=1

ψ_n^δk+k−1n^−k |t^α_n |^k=O(logm) asm→ ∞, then the seriesP

anλn is summableψ− |C, α;δ|_k.

If we take δ= 0, = 1 andψn=nin this theorem, then we get Theorem A.

4. We need the following lemmas for the proof of our theorem.

Lemma 1 ([5]). Ifσ > δ >0, then (10)

m

X

n=v+1

A^δ−1_n−v A^σ_n =

m

X

n=v+1

(n−v)^δ−1

n^σ =O(v^δ−σ) asm→ ∞.

Lemma 2 ([2]). Letλn →0 as n→ ∞. Suppose that there exists a sequence of numbers (Bn) which is δ-quasi monotone with PBnlogn is convergent and

|∆λn|≤|Bn|for alln, then

(11) |λ_n|logn=O(1) asn→ ∞.

Lemma 3 ([2]). Letα≥1. If (Bn) isδ-quasi monotone with P

n^αδnlogn <∞ and PB_nlognis convergent, then

(12) m^αBmlogm=O(1) as m→ ∞, (13)

∞

X

n=1

n^α|∆B_n|logn <∞.

Lemma 4 ([2]). Lett^α_n be the n-th Ces`aro mean of orderα, with α≥1, of the sequence (na_n) such thata_n≥0 for alln≥1 wheneverα >1. Ifn≥v, then

(14) |

v

X

p=1

A^α−1_n−ppap|≤A^α−1_n−vA^α_v |t^α_v |.

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5. Proof of the Theorem. Let (T_n^α) be the n-th (C, α), withα≥1, means of the sequence (nanλn). Then, by (3), we have

T_n^α = 1 A^α_n

n

X

v=1

A^α−1_n−vvavλv. (15)

Using Abel’s transformation, we get T_n^α = 1

A^α_n

n−1

X

v=1

∆λv v

X

p=1

A^α−1_n−ppap+ λ_n A^α_n

n

X

v=1

A^α−1_n−vvav

= 1

A^α_n

n−1

X

v=1

∆λ_v

v

X

p=1

A^α−1_n−ppa_p+λ_nt^α_n

= T_n,1^α +T_n,2^α , say. Since

|T_n,1^α +T_n,2^α |^k≤2^k(|T_n,1^α |^k+|T_n,2^α |^k), to complete the proof of the theorem, it is sufficient to show that

∞

X

n=1

ψ_n^δk+k−1n^−k |T_n,r^α |^k<∞forr= 1,2, by (6).

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Firstly, when k > 1, using Lemma 4 and after applying H¨older’s inequality with indices kandk⁰, where _k¹+_k¹0 = 1, we get that

m+1

X

n=2

ψ^δk+k−1_n n^−k|T_n,1^α |^k=

m+1

X

n=2

ψ^δk+k−1_n n^−k| 1 A^α_n

n−1

X

v=1

∆λv v

X

p=1

A^α−1_n−ppap|^k

≤

m+1

X

n=2

ψ^δk+k−1_n n^−k(A^α_n)^−k{

n−1

X

v=1

|Bv|A^α_vA^α−1_n−v|t^α_v |}^k

=O(1)

m+1

X

n=2

ψ^δk+k−1_n n^−k(A^α_n)^−k{

n−1

X

v=1

v^α|Bv|A^α−1_n−v|t^α_v |}^k

=O(1)

m+1

X

n=2

ψ^δk+k−1_n n^−kA^α_n

n−1

X

v=1

(v^α|Bv|)^kA^α−1_n−v|t^α_v |^k

× {

n−1

X

v=1

A^α−1_n−v A^α_n }^k−1

=O(1)

m

X

v=1

(v^α|B_v |)^k−1(v^α|B_v|)|t^α_v |^k

m+1

X

n=v

ψ_n^δk+k−1A^α−1_n−v n^kA^α_n

=O(1)

m

X

v=1

v^α|B_v||t^α_v |^k

m+1

X

n=v

ψ_n^δk+k−1n^−k(n−v)^α−1 n^α+

=O(1)

m

X

v=1

v^α|Bv||t^α_v |^k ψ_v^δk+k−1v^−k

m+1

X

n=v

(n−v)^α−1 n^α+

=O(1)

m

X

v=1

v^α|Bv||t^α_v |^k ψ_v^δk+k−1v^−k, by Lemma 1. Thus

m+1

X

n=2

ψ^δk+k−1_n n^−k|T_n,1^α |^k=O(1)

m−1

X

v=1

∆(v^α|Bv |)

v

X

p=1

ψ^δk+k−1_p p^−k|t^α_p |^k

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+O(1)m^α|Bm|

m

X

v=1

ψ^δk+k−1_v v^−k|t^α_v |^k

=O(1)

m−1

X

v=1

∆(v^α|Bv |) logv+O(1)m^α|Bm|logm

=O(1)

m−1

X

v=1

v^α|∆Bv|logv+O(1)

m−1

X

v=1

|∆(v^α)||Bv+1|logv +O(1)m^α|Bm|logm=O(1) asm→ ∞,

by virtue of the hypotheses of the Theorem and Lemma 3.

Again, since |λ_n|=O(1), we have that

m

X

n=1

ψ^δk+k−1_n n^−k|T_n,2^α |^k=

m

X

n=1

ψ_n^δk+k−1n^−k |λnt^α_n |^k

=

m

X

n=1

ψ_n^δk+k−1n^−k |λ_n|^k−1|λ_n||t^α_n |^k

=O(1)

m

X

n=1

ψ^δk+k−1_n n^−k|λn ||t^α_n|^k

=O(1)

m−1

X

n=1

∆|λn|

n

X

p=1

ψ_p^δk+k−1p^−k |t^α_p |^k

+O(1)|λ_m|

m

X

n=1

ψ^δk+k−1_n n^−k|t^α_n |^k

=O(1)

m−1

X

n=1

|∆λ_n|logn+O(1)|λ_m|logm

=O(1)

m−1

X

n=1

|Bn|logn+O(1)|λm|logm

=O(1) as m→ ∞,

by virtue of the hypotheses of the Theorem and Lemma 2.

Therefore, we get (16). This completes the proof of the Theorem.

References

[1] R. P. Boas, Quasi-positive sequences and trigonometric series,Proc. London Math. Soc.,14A (1965), 38–46.

[2] H. Bor, Absolute Ces`aro summability factors, Atti Sem. Math. Fis. Univ. Modena XLII (1994), 135–140.

[3] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley,Proc. Lond. Math. Soc.,7(1957), 113–141.

[4] E. Kogbetliantz, Sur les sèries absolument sommables par la mèthode des moyannes arithmètiques,Bull. Sci. Math.,49(1925), 234–256.

[5] W. T. Sulaiman, A study on a relation between two summability methods,Proc. Amer. Soc., 115(1992), 303–312..

Received April 11, 2000, in revised form July 19, 2000.

E-mail address: [email protected]

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Department of Mathematics Erciyes University

38039, Kayseri, Turkey