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volume 7, issue 4, article 135, 2006.

Received 18 March, 2006;

accepted 03 July, 2006.

Communicated by:H. Bor

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Journal of Inequalities in Pure and Applied Mathematics

A NOTE ON SUMMABILITY FACTORS

S.M. MAZHAR

Department of Mathematics and Computer Science Kuwait University

P.O. Box No. 5969, Kuwait - 13060.

EMail:sm_mazhar@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 081-06

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A Note on Summability Factors S.M. Mazhar

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J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006

http://jipam.vu.edu.au

Abstract

In this note we investigate the relation between two theorems proved by Bor [2,3] on|N, p¯ _n|_ksummability of an infinite series.

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

Key words: Absolute summability factors.

1. Introduction

LetP

a_n be a given infinite series with{s_n}as the sequence of its n-th partial sums. Let {pn}be a sequence of positive constants such thatPn = p0+p1 + p₂+· · ·+p_n −→ ∞asn −→ ∞.

Let

t_n= 1 P_n

n

X

ν=1

p_νs_ν.

The seriesP

a_nis said to be summable|N , p¯ _n|ifP∞

1 |t_n−tn−1| < ∞.It is said to be summable|N , p¯ _n|_k, k ≥1[1] if

(1.1)

∞

X

1

P_n p_n

k−1

|t_n−tn−1|^k <∞,

and bounded[ ¯N , p_n]_k, k≥1if

(1.2)

n

X

1

p_ν|s_ν|^k =O(P_n), n−→ ∞.

Concerning |N , p¯ _n|summability factors of P

a_n, T. Singh [6] proved the following theorem:

Theorem A. If the sequences{pn}and{λn}satisfy the conditions

(1.3)

∞

X

1

pn|λn|<∞,

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(1.4) P_n|∆λ_n| ≤Cp_n|λ_n|, C is a constant, and ifP

a_n is bounded[ ¯N , p_n]₁, thenP

a_nP_nλ_nis summable

|N , p¯ _n|.

Earlier in 1968 N. Singh [5] had obtained the following theorem.

Theorem B. IfP

a_nis bounded[ ¯N , p_n]₁and{λ_n}is a sequence satisfying the following conditions

(1.5)

∞

X

1

p_n|λ_n| P_n <∞,

(1.6) Pn

p_n∆λn =O(|λn|), thenP

a_nλ_nis summable|N , p¯ _n|.

In order to extend these theorems to the summability|N , p¯ _n|_k, k ≥ 1,Bor [2,3] proved the following theorems.

Theorem C. Under the conditions (1.2), (1.3) and (1.4), the seriesP

a_nP_nλ_n is summable|N , p¯ _n|_k, k ≥1.

Theorem D. If P

a_n is bounded [ ¯N , p_n]_k, k ≥ 1 and {λ_n}, is a sequence satisfying the conditions (1.4) and (1.5), thenP

a_nλ_nis summable|N , p¯ _n|_k.

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2. Results

In this note we propose to examine the relation between TheoremCand Theo- remD.

We recall that recently Sarigol and Ozturk [4] constructed an example to demonstrate that the hypotheses of TheoremAare not sufficient for the summability |N , p¯ _n|ofP

a_nP_nλ_n. They proved that Theorem Aholds true if we as- sume the additional condition

(2.1) pn+1 =O(pn).

From (1.4) we find that

∆λ_n λ_n

=

1−λ_n+1 λ_n

≤ Cp_n P_n , Hence

λ_n+1 λ_n

=

λ_n+1

λ_n −1 + 1

≤

1− λ_n+1 λ_n

+ 1

≤ Cp_n Pn

+ 1≤C.

Thus|λn+1| ≤C|λn|, and combining this with (2.1) we get (2.2) pn+1|λn+1| ≤Cpn|λn|.

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Clearly (2.1) and (1.4) imply (2.2). However (2.2) need not imply (2.1) or (1.4).

In view of

∆(P_nλ_n) =P_n∆λ_n−p_n+1λ_n+1

it is clear that if (2.2) holds, then the condition (1.4) is equivalent to the condition

(2.3) |∆(P_nλ_n)| ≤Cp_n|λ_n|.

It can be easily verified that a corrected version of TheoremAand TheoremC and also a slight generalization of the result of Sarigol and Ozturk fork = 1can be stated as

Theorem 2.1. Under the conditions (1.2), (1.3) (2.2) and (2.3) the seriesP

a_nP_nλ_n is summable|N , p¯ n|k, k ≥1

We now proceed to show that Theorem 2.1 holds good without condition (2.2).

Thus we have:

Theorem 2.2. Under the conditions (1.2), (1.3) and (2.3) the seriesP

a_nP_nλ_n is summable|N , p¯ _n|_k, k ≥1.

To prove Theorem2.2we first prove the following lemma.

Lemma 2.3. Under the conditions of Theorem2.2 (2.4)

m

X

1

p_n|λ_n||s_n|^k =O(1) as m−→ ∞.

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3. Proofs

Proof of Lemma2.3. In view of (1.3) and (2.3)

∞

X

1

|∆(λ_nP_n)| ≤C

∞

X

1

p_n|λ_n|<∞,

so it follows that{P_nλ_n} ∈BV and henceP_n|λ_n|=O(1).

Now

m

X

1

p_nλ_n||s_n|^k=

m−1

X

1

∆|λ_n|

n

X

ν=1

p_ν|s_ν|^k+|λ_m|

m

X

ν=1

p_ν|s_ν|^k

=O(1)

m−1

X

1

|∆λ_n|P_n+O(|λ_m|P_m)

=O(1)

m−1

X

1

|∆(P_nλ_n)|+p_n+1|λ_n+1|

!

+O(1)

=O(1)

m−1

X

1

p_n|λ_n|+O(1)

m

X

1

p_n+1|λ_n+1|+O(1)

=O(1).

Proof of Theorem2.2. LetTndenote then^th N , p¯ n

means of the seriesP

anPnλn.

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Then

T_n = 1 P_n

n

X

ν=0

p_ν

ν

X

r=0

a_rP_rλ_r

= 1 P_n

n

X

ν=0

(Pn−Pν−1)aνPνλν.

so that forn≥1

T_n−Tn−1 = p_n P_nPn−1

n

X

ν=1

Pν−1a_νP_νλ_ν

= p_n P_nP_n−1

n−1

X

ν=1

∆(Pν−1P_νλ_ν)s_ν+p_nλ_ns_n

=L₁+L₂, say.

Thus to prove the theorem it is sufficient to show that

∞

X

1

P_n p_n

^k−1

|L_ν|^k <∞, ν = 1,2.

Now

≤Cp_νP_ν|λ_ν|

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in view of (2.3). So

m+1

X

n=2

P_n p_n

k−1

|L1|^k=O(1)

m+1

X

n=2

p_n P_nP_n−1^k

n−1

X

ν=1

pνPν|λν||sν|

!k

=O(1)

m+1

X

n=2

p_n P_nP_n−1^k

n−1

X

ν=1

(P_ν|λ_ν|)^k|s_ν|^kp_ν

! _n−1 X

ν=1

p_ν

!^k−1

=O(1)

m+1

X

n=2

pn

P_nPn−1 n−1

X

ν=1

P_ν|λ_ν|p_ν|s_ν|^k

=O(1)

m

X

ν=1

pν|λ_ν|s_ν|^k=O(1)

in view of the lemma andP_n|λ_n|=O(1).

Also

m+1

X

1

P_n p_n

k−1

|L2|^k=O(1)

m+1

X

1

pn|λn|^k|sn|^kP_n^k−1

=O(1)

m+1

X

1

p_n|λ_n||s_n|^k

=O(1).

This proves Theorem2.2.

Thus a generalization of a corrected version of TheoremC is Theorem2.2.

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Writingλ_n=µ_nP_nthe conditions (1.5) and (1.4) become

(3.1)

∞

X

1

pn|µn|<∞,

(3.2) |∆(P_nµ_n)| ≤Cp_n|µ_n|, consequently TheoremDcan be stated as:

IfP

a_n is bounded[ ¯N , P_n]_k, k ≥1and{µ_n}is a sequence satisfying (3.1) and (3.2) thenP

a_nP_nµ_nis summable|N , p¯ _n|_k, k ≥1.

Thus TheoremDis the same as Theorem2.2which is a generalization of the corrected version of TheoremC.

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References

[1] H. BOR, On |N , p¯ _n|_k summability methods and|N , p¯ _n|_k summability factors of infinite series, Ph.D. Thesis (1982), Univ. of Ankara.

[2] H. BOR, On |N , p¯ _n|_k summability factors, Proc. Amer. Math. Soc., 94 (1985), 419–422.

[3] H. BOR, On |N , p¯ n|k summability factors of infinite series, Tamkang J.

Math., 16 (1985), 13–20.

[4] M. ALI SARIGOLANDE. OZTURK, A note on|N , p¯ _n|_ksummability fac- tors of infinite series, Indian J. Math., 34 (1992), 167–171.

[5] N. SINGH, On |N , p¯ _n| summability factors of infinite series, Indian J.

Math., 10 (1968), 19–24.

[6] T. SINGH, A note on|N , p¯ _n|summability factors of infinite series, J. Math.

Sci., 12-13 (1977-78), 25–28.