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volume 1, issue 2, article 18, 2000.

Received 14 April, 2000;

accepted 27 April, 2000.

Communicated by:L. Leindler

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Journal of Inequalities in Pure and Applied Mathematics

AN APPLICATION OF ALMOST INCREASING AND DELTA-QUASI-MONOTONE SEQUENCES

H. BOR

Department of Mathematics Erciyes University

Kayseri 38039, TURKEY EMail:bor@erciyes.edu.tr

2000c Victoria University ISSN (electronic): 1443-5756 009-00

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An Application of Almost Increasing and δ-Quasi-Monotone Sequences

H. Bor

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J. Ineq. Pure and Appl. Math. 1(2) Art. 18, 2000

http://jipam.vu.edu.au

Abstract

In this paper a general theorem on absolute weighted mean summability factors has been proved under weaker conditions by using an almost increasing and δ-quasi-monotone sequences.

2000 Mathematics Subject Classification:40D15, 40F05

Key words: Almost Increasing Sequences, Quasi-monotone Sequences, Absolute Summability Factors, Infinite Series

1. Introduction

A sequence(bn)of positive numbers is said to beδ-quasi-monotone, ifbn > 0 ultimately and∆b_n ≥ −δ_n, where(δ_n)is a sequence of positive numbers (see [2]). Let P

a_n be a given infinite series with (s_n) as the sequence of its n-th partial sums. By un and tn we denote the n-th (C,1)means of the sequence (s_n)and (na_n), respectively. The series P

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

∞

X

n=1

n^k−1|u_n−un−1|^k =

∞

X

n=1

1

n |t_n|^k <∞.

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

P_n =

n

X

v=0

p_v → ∞ as n → ∞, (P_−i =p_−i = 0, i≥1).

The sequence-to-sequence transformation

z_n = 1 P_n

n

X

v=0

p_vs_v defines the sequence(z_n)of the N , p¯ _n

mean of the sequence(s_n)generated by the sequence of coefficients (p_n) (see [6]). The series P

a_n is said to be summable

N , p¯ n

k,k ≥1, if (see [3])

∞

X

n=1

P_n p_n

k−1

|zn−zn−1|^k <∞.

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In the special case p_n = 1 for all values of n (resp. k = 1), then N , p¯ _n

k

summability is the same as|C,1|_k(resp.

N , p¯ _n

) summability. Also if we take pn = _n+1¹ , then

N , p¯ n

ksummability reduces to

N ,¯ _n+1¹

k summability.

Mazhar [7] has proved the following theorem for summability factors by usingδ-quasi-monotone sequences.

Theorem 1.1. Let λ_n → 0 asn → ∞. Suppose that there exists a sequence of numbers (A_n) such that it is δ-quasi-monotone with P

nδ_nlogn < ∞, PA_nlognis convergent and|∆λ_n| ≤ |A_n|for alln. If

m

X

n=1

1

n |t_n|^k =O(logm) as m→ ∞, then the seriesP

a_nλ_nis summable|C,1|_k,k ≥1.

Later on Bor [4] generalized Theorem1.1for a N , p¯ _n

ksummability method in the following form.

Theorem 1.2. Let λ_n → 0 asn → ∞and let (p_n) be a sequence of positive numbers such that

P_n =O(np_n) as n→ ∞.

Suppose that there exists a sequence of numbers (An) such that it is δ-quasi- monotone withP

nδ_nX_n <∞, P

A_nX_n is convergent and|∆λ_n| ≤ |A_n|for alln. If

m

X

n=1

p_n

P_n|t_n|^k =O(X_m) as m→ ∞, where(X_n)is a positive increasing sequence, then the seriesP

a_nλ_nis summable

N , p¯ n

k,k ≥1.

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It should be noted that if we takeX_n = lognandp_n = 1for all values ofn in Theorem1.2, then we get Theorem1.1.

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2. The Main Result.

Due to the restrictionP_n =O(np_n)on(p_n),no result forp_n= _n+1¹ can be de- duced from Theorem 1.2. Therefore the aim of this paper is to prove Theorem 1.2 under weaker conditions and in a more general form without this condi- tion. For this we need the concept of almost increasing sequence. A positive sequence(d_n)is said to be almost increasing if there exists a positive increasing sequence(c_n)and two positive constantsAandB such thatAc_n ≤ d_n ≤ Bc_n (see [1]). Obviously, every increasing sequence is almost increasing but the converse need not be true as can be seen from the exampled_n =ne⁽⁻¹⁾ⁿ. Since (X_n)is increasing in Theorem1.2, we are weakening the hypotheses of the theorem by replacing the increasing sequence with an almost increasing sequence.

Now, we shall prove the following theorem.

Theorem 2.1. Let(X_n)be an almost increasing sequence andλ_n →0asn→

∞. Suppose that there exists a sequence of numbers(A_n)such that it isδ-quasi- monotone withP

nδnXn <∞, P

AnXn is convergent and|∆λn| ≤ |An|for alln. If

m

X

n=1

1

n|λ_n| = O(1),

m

X

n=1

1

n|t_n|^k = O(X_m) as m → ∞

and m

X

n=1

p_n

P_n|tn|^k =O(Xm) as m→ ∞,

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then the seriesP

a_nλ_nis summable N , p¯ _n

k,k≥1.

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

Lemma 2.2. Under the conditions of the theorem, we have

|λn|Xn =O(1) as n→ ∞.

Proof. Sinceλ_n→0asn→ ∞, we have that

|λn|Xn=Xn

∞

X

v=n

∆λv

≤Xn

∞

X

v=n

|∆λv| ≤

∞

X

v=0

Xv|∆λv| ≤

∞

X

v=0

Xv|Av|<∞.

Hence|λ_n|X_n=O(1)asn → ∞.

Lemma 2.3. Let (Xn) be an almost increasing sequence. If(An)is δ-quasi- monotone withP

nδ_nX_n <∞,P

A_nX_nis convergent, then

nA_nX_n=O(1),

∞

X

n=1

nXn|∆An|<∞.

The proof of Lemma2.3 is similar to the proof of Theorem 1 and Theorem 2 of Boas [2, caseγ = 1], and hence is omitted.

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3. Proof of the Theorem

Proof of Theorem2.1. 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

Tn= 1 P_n

n

X

v=0

pv v

X

i=0

aiλi = 1 P_n

n

X

v=0

(Pn−Pv−1)avλv.

Then, forn≥1, we have

T_n−Tn−1 = p_n P_nP_n−1

n

X

v=1

Pv−1a_vλ_v = p_n P_nP_n−1

n

X

v=1

Pv−1λ_v v va_v. By Abel’s transformation, we get

T_n−Tn−1 = n+ 1 nPn

p_nt_nλ_n− p_n PnPn−1

n−1

X

v=1

p_vt_vλ_vv+ 1 v

+ p_n P_nPn−1

n−1

X

v=1

Pvtv∆λv

v+ 1

v + p_n P_nPn−1

n−1

X

v=1

Pvtvλv+1

1 v

= 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+|T_n,4|^k , to complete the proof of the theorem, it is enough to show that

∞

X

n=1

P_n p_n

k−1

|Tn,r|^k <∞ forr= 1,2,3,4.

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Sinceλ_nis bounded by the hypothesis, we have that

m

X

n=1

Pn

p_n k−1

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

m

X

n=1

pn

P_n|t_n|^k|λ_n| |λ_n|^k−1

= O(1)

m

X

n=1

p_n

P_n|tn|^k|λn|

= O(1)

m−1

X

n=1

|∆λ_n|

n

X

v=1

p_v

P_v |t_v|^k+O(1)|λ_m|

m

X

n=1

p_n P_n |t_n|^k

= O(1)

m−1

X

n=1

|∆λ_n|X_n+O(1)|λ_m|X_m

= O(1)

m−1

X

n=1

|A_n|X_n+O(1)|λ_m|X_m

= O(1) as m→ ∞,

by virtue of the hypotheses of the theorem and Lemma2.2.

Now, whenk > 1, applying Hölder’s inequality, as inT_n,1, we have that

m+1

X

n=2

P_n p_n

k−1

|T_n,2|^k

=O(1)

m+1

X

n=2

pn

P_nPn−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

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= O(1)

m

X

v=1

p_v|t_v|^k|λ_v| |λ_v|^k−1

m+1

X

n=v+1

p_n P_nPn−1

= O(1)

m

X

v=1

|λ_v| p_v

P_v |t_v|^k=O(1) as m → ∞.

Again we have that,

m+1

X

n=2

P_n p_n

^k−1

|T_n,3|^k

=O(1)

m+1

X

n=2

p_n P_nPn−1

n−1

X

v=1

P_v|t_v|^k|A_v| × ( 1

Pn−1 n−1

X

v=1

P_v|A_v| )^k−1

=O(1)

m

X

v=1

P_v|t_v|^k|A_v|

m+1

X

n=v+1

p_n PnPn−1

=O(1)

m

X

v=1

|t_v|^k|A_v|

=O(1)

m

X

v=1

v|A_v|1 v |t_v|^k

=O(1)

m−1

X

v=1

∆ (v|A_v|)

v

X

i=1

1

i |t_i|^k+O(1)m|A_m|

m

X

v=1

1 v |t_v|^k

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=O(1)

m−1

X

v=1

|∆ (v|A_v|)|X_v+O(1)m|A_m|X_m

=O(1)

m−1

X

v=1

v|∆A_v|X_v +O(1)

m−1

X

v=1

|A_v+1|X_v+1+O(1)m|A_m|X_m

=O(1) as m→ ∞,

in view of the hypotheses of Theorem2.1and Lemma2.3.

Finally, we get that

m+1

X

n=2

Pn

p_n k−1

|T_n,4|^k

≤

m+1

X

n=2

p_n P_nPn−1

n−1

X

v=1

Pv|tv|^k|λv+1|1 v ×

( 1 Pn−1

n−1

X

v=1

Pv|λv+1|1 v

)^k−1

=O(1)

m

X

v=1

P_v|t_v|^k|λ_v+1|1 v

m+1

X

n=v+1

p_n P_nPn−1

=O(1)

m

X

v=1

|λ_v+1|1 v |t_v|^k

=O(1)

m−1

X

v=1

∆|λ_v+1|

v

X

i=1

1

i |t_i|^k+O(1)|λ_m+1|

m

X

v=1

1 v |t_v|^k

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=O(1)

m−1

X

v=1

|∆λ_v+1|X_v+1+O(1)|λ_m+1|X_m+1

=O(1)

m−1

X

v=1

|λ_v+1|X_v+1+O(1)|λ_m+1|X_m+1

=O(1) as m→ ∞,

by virtue of the hypotheses of Theorem2.1and Lemma2.2.

Therefore we get

m

X

n=1

P_n p_n

k−1

|T_n,r|^k=O(1) as m → ∞, forr = 1,2,3,4.

This completes the proof of the theorem.

If we takep_n = 1for all values of n(resp. p_n = _n+1¹ ), then we get a result concerning the|C,1|_k (resp.

N ,¯ _n+1¹

k) summability factors.

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References

[1] L.S. ALJANCICANDD. ARANDELOVIC,O- regularly varying functions, Publ. Inst. Math., 22 (1977), 5–22.

[2] R. P. BOAS, Quasi positive sequences and trigonometric series, Proc. Lon- don Math. Soc., 14A (1965), 38–46.

[3] H. BOR, On two summability methods, Math. Proc.Camb. Philos. Soc., 97 (1985), 147–149.

[4] H. BOR, On quasi monotone sequences and their applications, Bull. Austral.

Math. Soc., 43 (1991), 187–192.

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

[6] G. H. HARDY, Divergent Series, Oxford Univ. Press, 1949.

[7] S. M. MAZHAR, On generalized quasi-convex sequence and its applica- tions, Indian J. Pure Appl. Math., 8 (1977), 784–790.