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volume 4, issue 4, article 66, 2003.

Received 07 April, 2003;

accepted 30 April, 2003.

Communicated by:L. Leindler

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Journal of Inequalities in Pure and Applied Mathematics

INCLUSION THEOREMS FOR ABSOLUTE SUMMABILITY METHODS

H. S. ÖZARSLAN

Department of Mathematics, Erciyes University,

38039 Kayseri, Turkey.

EMail:seyhan@erciyes.edu.tr

c

2000Victoria University ISSN (electronic): 1443-5756 046-03

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Inclusion Theorems for Absolute Summability Methods

H. S. Özarslan

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J. Ineq. Pure and Appl. Math. 4(4) Art. 66, 2003

http://jipam.vu.edu.au

Abstract

In this paper we have proved two theorems concerning an inclusion between two absolute summability methods by using any absolute summability factor.

2000 Mathematics Subject Classification:40D15, 40F05, 40G99 Key words: Absolute summability, Summability factors, Infinite series.

1. Introduction

Let P

a_n be a given infinite series with partial sums(s_n), andr_n = na_n. By u_n and t_n we denote the n-th (C,1) means of the sequences (s_n) and (r_n), respectively. The seriesP

a_nis said to be summable|C,1|_k, k ≥1, if (see [4])

(1.1)

∞

X

n=1

n^k−1|u_n−un−1|^k <∞.

But since t_n = n(u_n−un−1)(see [7]), the condition (1.1) can also be written as

(1.2)

∞

X

n=1

1

n|t_n|^k <∞.

The seriesP

a_n is said to be summable|C,1;δ|_k k ≥ 1and δ ≥ 0, if (see [5])

(1.3)

∞

X

n=1

n^δk−1|tn|^k <∞.

If we takeδ = 0, then|C,1;δ|_ksummability is the same as|C,1|_ksummability.

Let(p_n)be a sequence of positive numbers such that (1.4) Pn =

n

X

v=0

pv → ∞ as n→ ∞, (P−i =p−i = 0, i≥1).

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The sequence-to-sequence transformation

(1.5) T_n = 1

P_n

n

X

v=0

p_vs_v

defines the sequence(T_n)of the( ¯N , p_n)mean of the sequence(s_n), generated by the sequence of coefficients(p_n)(see [6]).

The seriesP

a_nis said to be summable N , p¯ _n

k, k ≥1,if (see [1]) (1.6)

∞

X

n=1

P_n p_n

^k−1

|∆Tn−1|^k <∞

and it is said to be summable

N , p¯ _n;δ

k,k ≥1andδ ≥0,if (see [3]) (1.7)

∞

X

n=1

P_n p_n

δk+k−1

|∆Tn−1|^k <∞,

where

(1.8) ∆Tn−1 =− pn

P_nPn−1 n

X

v=1

Pv−1a_v, n≥1.

In the special case when δ = 0 (resp. p_n = 1for all values of n)

N , p¯ _n;δ k

summability is the same as N , p¯ _n

k(resp. |C,1;δ|_k) summability.

Concerning inclusion relations between|C,1|_k and N , p¯ _n

ksummabilities, the following theorems are known.

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Theorem 1.1. ([1]).Letk ≥ 1and let(p_n)be a sequence of positive numbers such that asn → ∞

(1.9) (i) P_n =O(np_n), (ii) np_n =O(P_n).

If the seriesP

a_nis summable|C,1|_k,then it is also summable N , p¯ _n

k. Theorem 1.2. ([2]). Let k ≥ 1 and let (p_n) be a sequence of positive num- bers such that condition (1.9) of Theorem1.1is satisfied. If the series P

a_nis summable

N , p¯ _n

k,then it is also summable|C,1|_k.

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2. The Main Result

The aim of this paper is to generalize the above theorems for |C,1;δ|_k and

N , p¯ _n;δ

ksummabilities, by using a summability factors. Now, we shall prove the following theorems.

Theorem 2.1. Let k ≥ 1and0 ≤ δk < 1. Let(p_n)be a sequence of positive numbers such thatP_n =O(np_n)and

(2.1)

∞

X

n=v+1

P_n p_n

δk−1

1 Pn−1

=O (

P_v p_v

δk

1 P_v

) . Let P

a_n be summable |C,1;δ|_k. Then P

a_nλ_n is summable

N , p¯ _n;δ k, if (λ_n)satisfies the following conditions:

(2.2) n∆λ_n =O

P_n npn

^1−δk_k ,

(2.3) λ_n=O

P_n np_n

¹_k .

Theorem 2.2. Let k ≥ 1and0 ≤ δk < 1. Let(p_n)be a sequence of positive numbers such that np_n = O(P_n) and satisfies the condition (2.1). Let P

a_n be summable

N , p¯ _n;δ

k.ThenP

a_nλ_nis summable|C,1;δ|_k,if(λ_n)satisfies the following conditions:

(2.4) n∆λ_n =O

np_n P_n

^1−δk_k

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(2.5) λ_n=O

np_n P_n

¹_k .

Remark 2.1. It may be noted that, if we takeλ_n = 1andδ= 0in Theorem2.1 and Theorem 2.2, then we get Theorem 1.1 and Theorem1.2, respectively. In this case condition (2.1) reduces to

m+1

X

n=v+1

p_n P_nPn−1

=

m+1

X

n=v+1

1 Pn−1

− 1 P_n

=O 1

P_v

as m → ∞,

which always holds.

Proof of Theorem2.1. Since

t_n= 1 n+ 1

n

X

v=1

va_v

we have that

a_n= n+ 1

n t_n−tn−1. Let(T_n)denote the( ¯N , p_n)mean of the seriesP

a_nλ_n. Then, by definition and changing the order of summation, we have

T_n= 1 P_n

n

X

v=0

p_v

v

X

i=0

a_iλ_i = 1 P_n

n

X

v=0

(P_n−P_v−1)a_vλ_v.

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Then, forn≥1, we have

T_n−Tn−1 = p_n PnPn−1

n

X

v=1

Pv−1a_vλ_v.

By Abel’s transformation, we have T_n−T_n−1 = p_n

P_nPn−1 n−1

X

v=1

P_v∆λ_vv+ 1

v t_v− p_n P_nPn−1

n−1

X

v=1

p_vλ_vv+ 1 v t_v

+ p_n P_nPn−1

n−1

X

v=1

P_v

v λ_v+1t_v+ p_n

P_nλ_nn+ 1 n t_n

=T_n,1+T_n,2+T_n,3+T_n,4, say.

Since

|Tn,1 +Tn,2+Tn,3+Tn,4|^k ≤4^k(|Tn,1|^k+|Tn,2|^k+|Tn,3|^k+|Tn,4|^k),

to complete the proof of the theorem, it is enough to show that (2.6)

∞

X

n=1

P_n p_n

δk+k−1

|T_n,r|^k <∞ for r = 1,2,3,4.

Now, when k > 1, applying Hölder’s inequality with indicesk andk⁰, where

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k +_k¹0 = 1, we get

m+1

X

n=2

P_n p_n

δk+k−1

|T_n,1|^k

≤

m+1

X

n=2

P_n pn

δk−1

1 P_n−1^k

n−1

X

v=1

P_vv+ 1

v |t_v| |∆λ_v|

!k

=O(1)

m+1

X

n=2

P_n p_n

^δk−1 1 P_n−1^k

n−1

X

v=1

|t_v| |v∆λ_v| P_v vp_vp_v

!k

=O(1)

m+1

X

n=2

P_n p_n

δk−1

1 P_n−1^k

n−1

X

v=1

|t_v| |v∆λ_v|p_v

!k

=O(1)

m+1

X

n=2

P_n p_n

δk−1

1 Pn−1

n−1

X

v=1

|t_v|^k|v∆λ_v|^kp_v 1 Pn−1

n−1

X

v=1

p_v

!^k−1

=O(1)

m

X

v=1

|t_v|^k|v∆λ_v|^kp_v

m+1

X

n=v+1

P_n pn

^δk−1 1 Pn−1

=O(1)

m

X

v=1

|t_v|^k|v∆λ_v|^k P_v

p_v δk−1

=O(1)

m

X

v=1

v^δk−1|t_v|^k =O(1) as m → ∞.

by virtue of the hypotheses of Theorem2.1.

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Again using Hölder’s inequality,

m+1

X

n=2

P_n pn

δk+k−1

|T_n,2|^k

=O(1)

m+1

X

n=2

P_n pn

^δk−1 1 Pn−1

n−1

X

v=1

p_v|t_v|^k|λ_v|^k 1 Pn−1

n−1

X

v=1

p_v

!k−1

=O(1)

m

X

v=1

pv|tv|^k|λv|^k

m+1

X

n=v+1

P_n p_n

δk−1

1 Pn−1

=O(1)

m

X

v=1

Pv

p_v δk−1

|t_v|^k|λ_v|^k

=O(1)

m

X

v=1

v^δk−1|t_v|^k|λ_v|^k

=O(1)

m

X

v=1

v^δk−1|t_v|^k=O(1) as m → ∞, by virtue of the hypotheses of Theorem2.1.

Also

m+1

X

n=2

Pn

p_n

δk+k−1

|T_n,3|^k

≤

m+1

X

n=2

P_n p_n

δk−1

1 P_n−1^k

n−1

X

v=1

Pv

|λ_v+1| v |tv|

!k

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=O(1)

m+1

X

n=2

P_n p_n

δk−1

1 P_n−1^k

n−1

X

v=1

vp_v|λ_v+1| v |t_v|

!k

=O(1)

m+1

X

n=2

Pn

p_n δk−1

1 Pn−1

n−1

X

v=1

p_v|λ_v+1|^k|t_v|^k 1 Pn−1

n

X

v=1

p_v

!k−1

=O(1)

m

X

v=1

p_v|λ_v+1|^k|t_v|^k

m+1

X

n=v+1

P_n p_n

^δk−1 1 P_n−1

=O(1)

m

X

v=1

|t_v|^k P_v

pv

δk−1

|λ_v+1|^k

=O(1)

m

X

v=1

v^δk−1|t_v|^k=O(1) as m→ ∞, by virtue of the hypotheses of Theorem2.1.

Lastly

m

X

n=1

P_n p_n

δk+k−1

|T_n,4|^k =O(1)

m

X

n=1

P_n p_n

δk−1

|λ_n|^k|t_n|^k

=O(1)

m

X

n=1

P_n npn

δk−1

|λ_n|^k|t_n|^kn^δk−1

=O(1)

m

X

n=1

n^δk−1|t_n|^k =O(1) as m → ∞, by virtue of the hypotheses of Theorem2.1.

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This completes the proof of Theorem2.1.

Proof of Theorem2.2. Let (T_n) denotes the( ¯N , p_n) mean of the series P a_n. We have

Tn= 1 P_n

n

X

v=0

pvsv = 1 P_n

n

X

v=0

(Pn−Pv−1)av. Since

T_n−Tn−1 = pn

P_nPn−1 n

X

v=0

Pv−1a_v,

we have that

(2.7) a_n = −P_n

p_n ∆Tn−1+ Pn−2

p_n−1∆Tn−2. Let

tn= 1 n+ 1

n

X

v=1

vavλv.

By using (2.7) we get tn= 1

n+ 1

n

X

v=1

v

−P_v

p_v ∆Tv−1+ P_v−2 pv−1

∆Tv−2

λv

= 1 n+ 1

n−1

X

v=1

(−v)P_v

p_v∆Tv−1λ_v− nP_nλ_n

(n+ 1)p_n∆Tn−1

+ 1 n+ 1

n

X

v=1

vP_v−2 pv−1

∆Tv−2λv

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= 1 n+ 1

n−1

X

v=1

(−v)P_v

p_v∆Tv−1λ_v

+ 1 n+ 1

n−1

X

v=1

(v+ 1)Pv−1

p_v ∆Tv−1λ_v+1− nP_nλ_n

(n+ 1)p_n∆Tn−1

= 1 n+ 1

n−1

X

v=1

∆T_v−1

p_v {−vλ_vP_v + (v+ 1)λ_v+1P_v−1} − nP_nλ_n

(n+ 1)p_n∆T_n−1

= 1 n+ 1

n−1

X

v=1

∆Tv−1

p_v {−vλ_vP_v + (v+ 1)λ_v+1(P_v −p_v)} − nPnλn

(n+ 1)p_n∆Tn−1

= 1 n+ 1

n−1

X

v=1

∆Tv−1

p_v {−vλ_vP_v + (v+ 1)λ_v+1P_v−(v+ 1)λ_v+1p_v}

− nP_nλ_n

(n+ 1)p_n∆Tn−1

= 1 n+ 1

n−1

X

v=1

∆Tv−1

p_v {−(∆vλ_v)P_v−(v+ 1)λ_v+1p_v} − nPnλn

(n+ 1)p_n∆Tn−1

= 1 n+ 1

n−1

X

v=1

−P_v

p_v ∆Tv−1{v∆λ_v−λ_v+1} − 1 n+ 1

n−1

X

v=1

∆Tv−1(v+ 1)λ_v+1

− nP_nλ_n

(n+ 1)p_n∆Tn−1

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=− 1 n+ 1

n−1

X

v=1

vP_v

p_v ∆λ_v∆Tv−1+ 1 n+ 1

n−1

X

v=1

P_v

p_v∆Tv−1λ_v+1

− 1

n+ 1

n−1

X

v=1

(v+ 1)λ_v+1∆Tv−1− nP_nλ_n

(n+ 1)p_n∆Tn−1

=t_n,1+t_n,2+t_n,3+t_n,4, say.

Since

|t_n,1+t_n,2 +t_n,3+t_n,4|^k ≤4^k(|t_n,1|^k+|t_n,2|^k+| |t_n,3|^k+|t4|^k), to complete the proof of Theorem2.2, it is enough to show that

∞

X

n=1

n^δk−1|t_n,r|^k <∞ f or r= 1,2,3,4.

Now, when k > 1 applying Hölder’s inequality with indicesk and k⁰, where

1

k +_k¹0 = 1, we have that

m+1

X

n=2

n^δk−1|t_n,1|^k ≤

m+1

X

n=2

n^δk−1 1 n^k

n−1

X

v=1

P_v

p_vv|∆λ_v| |∆T_v−1|

!k

≤

m+1

X

n=2

1 n^2−δk

n−1

X

v=1

P_v pv

k

|∆λ_v|^kv^k|∆Tv−1|^k 1 n

n−1

X

v=1

1

!k−1

≤

m+1

X

n=2

1 n^2−δk

n−1

X

v=1

P_v p_v

k

|∆λv|^kv^k|∆Tv−1|^k

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=O(1)

m

X

v=1

P_v p_v

k

|∆λ_v|^kv^k|∆Tv−1|^k

m+1

X

n=v+1

1 n^2−δk

=O(1)

m

X

v=1

P_v p_v

k

|∆λ_v|^kv^k|∆Tv−1|^k 1 v^1−δk

=O(1)

m

X

v=1

P_v p_v

δk+k−1

|∆Tv−1|^k=O(1) as m→ ∞, by virtue of the hypotheses of Theorem2.2.

Again using Hölder’s inequality,

m+1

X

n=2

n^δk−1|t_n,2|^k ≤

m+1

X

n=2

n^δk−1−k

n−1

X

v=1

|λ_v+1|P_v pv

|∆T_v−1|

!k

≤

m+1

X

n=2

1 n^2−δk

n−1

X

v=1

P_v pv

k

|λ_v+1|^k|∆Tv−1|^k 1 n

n−1

X

v=1

1

!k−1

=O(1)

m

X

v=1

P_v p_v

k

|λv+1|^k|∆Tv−1|^k

m+1

X

n=v+1

1 n^2−δk

=O(1)

m

X

v=1

Pv

p_v k

|λ_v+1|^k|∆Tv−1|^k 1 v^1−δk

=O(1)

m

X

v=1

P_v p_v

)^δk+k−1|∆T_v−1|^k

=O(1) as m→ ∞,

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Also, we have that

m+1

X

n=2

n^δk−1|t_n,3|^k ≤

m+1

X

n=2

n^δk−1−k

n−1

X

v=1

(v+ 1)|λ_v+1| |∆T_v−1|

!k

=O(1)

m+1

X

n=2

n^δk−1−k

n−1

X

v=1

v|λ_v+1| |∆Tv−1|

!k

=O(1)

m+1

X

n=2

1 n^2−δk

n−1

X

v=1

v^k|λ_v+1|^k|∆Tv−1|^k 1 n

n−1

X

v=1

1

!^k−1

=O(1)

m+1

X

n=2

1 n^2−δk

n−1

X

v=1

v^k|λ_v+1|^k|∆T_v−1|^k

=O(1)

m

X

v=1

v^k|λ_v+1|^k|∆Tv−1|^k

m+1

X

n=v+1

1 n^2−δk

=O(1)

m

X

v=1

v^δk+k−1|∆Tv−1|^k

=O(1)

m

X

v=1

Pv

p_v

δk+k−1

|∆Tv−1|^k

=O(1) as m→ ∞, by virtue of the hypotheses of Theorem2.2.

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Finally, we have that

m

X

n=1

n^δk−1|t_n,4|^k =O(1)

m

X

n=1

Pn

p_n k

n^δk−1|λ_n|^k|∆Tn−1|^k

=O(1)

m

X

n=1

Pn

p_n

δk+k−1

|∆Tn−1|^k

=O(1) as m→ ∞, by virtue of the hypotheses of Theorem2.2.

Therefore, we get that

m

X

n=1

n^δk−1|t_n,r|^k =O(1) as m→ ∞, for r= 1,2,3,4.

This completes the proof of Theorem2.2.

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References

[1] H. BOR, On two summability methods, Math. Proc. Cambridge Phil. Soc., 97 (1985), 147–149.

[2] H. BOR, A note on two summability methods, Proc. Amer. Math. Soc., 98 (1986), 81–84.

[3] H. BOR, On the local property of

N , p¯ n;δ

k summability of factored Fourier series, J. Math. Anal. Appl., 179 (1993), 646–649.

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

[5] T.M. FLETT, Some more theorems concerning the absolute summability of Fourier series, Proc. London Math. Soc., 8 (1958), 357–387.

[6] G.H. HARDY, Divergent Series, Oxford University Press., Oxford, (1949).

[7] E. KOGBETLIANTZ, Sur les series absolument sommables par la methode des moyannes aritmetiques, Bull. Sci. Math., 49(2) (1925), 234–256.