A GENERAL NOTE ON INCREASING SEQUENCES

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Increasing Sequences Hüseyin Bor vol. 8, iss. 3, art. 82, 2007

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A GENERAL NOTE ON INCREASING SEQUENCES

HÜSEY˙IN BOR

Department of Mathematics Erciyes University

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

Received: 23 December, 2006

Accepted: 08 August, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 40D15, 40F05, 40G99.

Key words: Absolute summability, Summability factors, Almost and power increasing sequences, Infinite series.

Abstract: In the present paper, a general theorem on |N , p¯ n|_k summability factors of infinite series has been proved under more weaker conditions. Also we have obtained a new result concerning the|C,1|_ksummability factors.

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1. Introduction

A positive sequence (b_n) is said to be almost increasing if there exists a positive increasing sequence(c_n)and two positive constantsAandBsuch thatAc_n ≤b_n ≤ Bc_n (see [1]). We denote byBVO the expression BV ∩ CO, whereCO and BV are the set of all null sequences and the set of all sequences with bounded variation, respectively. LetP

anbe a given infinite series with partial sums (sn). We denote byu^α_n andt^α_n then-th Cesàro means of orderα, withα >−1, of the sequences(s_n) and(na_n), respectively, i.e.,

(1.1) u^α_n = 1

A^α_n

n

X

v=0

A^α−1_n−vs_v,

(1.2) t^α_n = 1

A^α_n

n

X

v=1

A^α−1_n−vva_v, where

(1.3) A^α_n =O(n^α), α >−1, A^α₀ = 1 and A^α_−n = 0 for n >0.

The seriesP

a_nis said to be summable|C, α|_k,k ≥1, if (see [6,8]) (1.4)

∞

X

n=1

n^k−1

u^α_n−u^α_n−1

k=

∞

X

n=1

|t^α_n|^k n <∞.

If we takeα= 1, then we get|C,1|_ksummability.

Let(p_n)be a sequence of positive numbers such that (1.5) P_n =

n

X

v=0

p_v → ∞ as n→ ∞, (P−i =p−i = 0, i≥1).

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The sequence-to-sequence transformation

(1.6) σ_n= 1

P_n

n

X

v=0

p_vs_v

defines the sequence (σn) of the Riesz mean or simply the ( ¯N , pn) mean of the sequence(sn), generated by the sequence of coefficients(pn) (see [7]). The series Pa_nis said to be summable

N , p¯ _n

k, k ≥1,if (see [2,3]) (1.7)

∞

X

n=1

(P_n/p_n)^k−1|∆σn−1|^k <∞,

where

(1.8) ∆σn−1 =− p_n

PnPn−1 n

X

v=1

Pv−1a_v, n≥1.

In the special casep_n = 1 for all values of n, N , p¯ _n

k summability is the same as

|C,1|_ksummability.

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2. Known Results

Mishra and Srivastava [10] have proved the following theorem concerning the N , p¯ _n

summability factors.

Theorem A. Let(X_n) be a positive non-decreasing sequence and let there be se- quences(βn)and(λn)such that

(2.1) |∆λ_n| ≤β_n,

(2.2) βn →0 as n → ∞,

(2.3)

∞

X

n=1

n|∆β_n|X_n <∞,

(2.4) |λ_n|X_n=O(1).

If (2.5)

n

X

v=1

|s_v|

v =O(X_n) as n → ∞

and(p_n)is a sequence such that

(2.6) P_n=O(np_n),

(2.7) P_n∆p_n =O(p_np_n+1),

then the seriesP∞

n=1anPnλn

npn is summable N , p¯ n

.

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Later on Bor [4] generalized TheoremAfor N , p¯ _n

ksummability in the following form.

Theorem B. Let (X_n) be a positive non-decreasing sequence and the sequences (β_n)and(λ_n)are such that conditions (2.1) – (2.7) of TheoremAare satisfied with the condition (2.5) replaced by:

(2.8)

n

X

v=1

|s_v|^k

v =O(X_n) as n→ ∞.

Then the seriesP∞

n=1a_n^P_npⁿ^λⁿ

n is summable N , p¯ _n

k, k ≥1.

It may be noticed that if we takek = 1, then we get TheoremA.

Quite recently Bor [5] has proved TheoremBunder weaker conditions by taking an almost increasing sequence instead of a positive non-decreasing sequence.

Theorem C. Let (X_n) be an almost increasing sequence. If the conditions (2.1) – (2.4) and (2.6) – (2.8) are satisfied, then the series P∞

n is summable

N , p¯ _n

_k,k ≥1.

Remark 1. It should be noted that, under the conditions of Theorem B, (λ_n) is bounded and∆λ_n =O(1/n)(see [4]).

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3. Main Result

The aim of this paper is to prove Theorem C under weaker conditions. For this we need the concept of quasiβ-power increasing sequences. A positive sequence (γ_n) is said to be a quasi β-power increasing sequence if there exists a constant K =K(β, γ)≥1such that

(3.1) Kn^βγn≥m^βγm

holds for alln≥m ≥1. It should be noted that almost every increasing sequence is a quasiβ-power increasing sequence for any nonnegativeβ, but the converse need not be true as can be seen by taking the example, sayγ_n=n^−β forβ >0.

Now we shall prove the following theorem.

Theorem 3.1. Let(X_n)be a quasiβ-power increasing sequence for some0< β <

1. If the conditions (2.1) – (2.4), (2.6) – (2.8) and

(3.2) (λ_n)∈ BVO

are satisfied, then the seriesP∞

n is summable N , p¯ _n

k,k≥1.

It should be noted that if we take(X_n)as an almost increasing sequence, then we get TheoremC. In this case, condition (3.2) is not needed.

We require the following lemma for the proof of Theorem3.1.

Lemma 3.2 ([9]). Except for the condition (3.2), under the conditions on(X_n),(β_n) and (λ_n) as taken in the statement of Theorem 3.1, the following conditions hold, when (2.3) is satisfied:

(3.3) nXnβn =O(1),

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(3.4)

∞

X

n=1

β_nX_n<∞.

Proof of Theorem3.1. Let(T_n)denote the( ¯N , p_n)mean of the seriesP∞ n=1

anPnλn

npn . Then, by definition, we have

(3.5) T_n= 1 P_n

n

X

v=1

p_v

v

X

r=1

a_rP_rλ_r rp_r = 1

P_n

n

X

v=1

(P_n−Pv−1)a_vP_vλ_v vp_v , and thus

(3.6) T_n−T_n−1 = p_n P_nPn−1

n

X

v=1

Pv−1P_va_vλ_v

vp_v , n ≥1.

Using Abel’s transformation, we get T_n−T_n−1 = p_n

P_nPn−1 n

X

v=1

s_v∆

Pv−1P_vλ_v vp_v

+λ_ns_n n

= snλn

n + pn

P_nPn−1 n−1

X

v=1

s_vPv+1Pv∆λv

(v+ 1)p_v+1

+ p_n P_nP_n−1

n−1

X

v=1

P_vs_vλ_v∆ P_v

vp_v

− p_n P_nP_n−1

n−1

X

v=1

s_vP_vλ_v1 v

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

To prove Theorem3.1, by Minkowski’s inequality, it is sufficient to show that (3.7)

∞

X

n=1

Pn

p_n k−1

|T_n,r|^k <∞, for r = 1,2,3,4.

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Firstly by using Abel’s transformation, we have

m

X

n=1

P_n p_n

k−1

|T_n,1|^k =

m

X

n=1

P_n np_n

k−1

|λ_n|^k−1|λ_n||s_n|^k n

=O(1)

m

X

n=1

|λ_n|| |s_n| |^k n

=O(1)

m−1

X

n=1

∆|λ_n|

n

X

v=1

|s_v|^k

v +O(1)|λ_m|

m

X

n=1

|s_n|^k n

=O(1)

m−1

X

n=1

|∆λ_n|X_n+O(1)|λ_m|X_m

=O(1)

m−1

X

n=1

β_nX_n+O(1)|λ_m|X_m =O(1) as m → ∞, by virtue of the hypotheses of Theorem3.1and Lemma3.2.

Now, using the fact that P_v+1 = O((v + 1)p_v+1), by (2.6), and then applying Hölder’s inequality, we have

m+1

X

n=2

P_n p_n

k−1

|T_n,2|^k=O(1)

m+1

X

n=2

p_n P_nP_n−1^k

n−1

X

v=1

P_vs_v∆λ_v

k

=O(1)

m+1

X

n=2

p_n P_nP_n−1^k

(_n−1 X

v=1

P_v

p_v |s_v|p_v|∆λ_v| )k

=O(1)

m+1

X

n=2

pn

P_nPn−1 n−1

X

v=1

Pv

p_v k

|s_v|^kp_v|∆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 p_v

k

|s_v|^kp_v|∆λ_v|^k

m+1

X

n=v+1

p_n P_nPn−1

=O(1)

m

X

v=1

P_v|∆λ_v| p_v

k−1

|sv|^k|∆λv|

=O(1)

m

X

v=1

|sv|^k|∆λv| P_v

vp_v k−1

=O(1)

m

X

v=1

vβ_v|s_v|^k v

=O(1)

m−1

X

v=1

∆(vβv)

v

X

r=1

|s_r|^k

r +O(1)mβm m

X

v=1

|s_v|^k v

=O(1)

m−1

X

v=1

|∆(vβ_v)|X_v+O(1)mβ_mX_m

=O(1)

m−1

X

v=1

v|∆β_v|X_v+O(1)

m−1

X

v=1

|β_v|X_v+O(1)mβ_mX_m

=O(1)

asm → ∞, in view of the hypotheses of Theorem3.1and Lemma3.2.

Again, since∆(_vp^P^v

v) =O(¹_v),by (2.6) and (2.7) (see [10]), as inT_n,1we have

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_nP_n−1^k

(_n−1 X

v=1

P_v|s_v| |λ_v|1 v

)^k

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=O(1)

m+1

X

n=2

p_n P_nP_n−1^k

(_n−1 X

v=1

P_v p_v

p_v|s_v| |λ_v|1 v

)k

=O(1)

m+1

X

n=2

p_n P_nPn−1

n−1

X

v=1

P_v vp_v

k

p_v|s_v|^k|λ_v|^k ( 1

Pn−1 n−1

X

v=1

p_v )^k−1

=O(1)

m

X

v=1

P_v vp_v

k

|s_v|^kp_v|λ_v|^k

m+1

X

n=v+1

p_n P_nP_n−1

=O(1)

m

X

v=1

P_v vpv

k

p_v|s_v|^k|λ_v|^k 1 Pv

.v v

=O(1)

m

X

v=1

P_v vpv

k−1

|λ_v|^k−1|λ_v||s_v|^k v

=O(1)

m

X

v=1

|λ_v||s_v|^k v

=O(1)

m−1

X

v=1

X_vβ_v +O(1)X_m|λ_m|=O(1) as m→ ∞.

Finally, using Hölder’s inequality, as inT_n,3 we have

m+1

X

n=2

P_n p_n

k−1

|T_n,4 |^k =

m+1

X

n=2

p_n P_nP_n−1^k

n−1

X

v=1

s_vP_v v λ_v

k

=

m+1

X

n=2

p_n P_nP_n−1^k

n−1

X

v=1

s_v P_v vpv

p_vλ_v

k

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≤

m+1

X

n=2

p_n P_nPn−1

n−1

X

v=1

|s_v|^k P_v

vp_v k

p_v|λ_v|^k 1 Pn−1

n−1

X

v=1

p_v

!^k−1

=O(1)

m

X

v=1

P_v vp_v

k

|s_v|^kp_v|λ_v|^k 1 P_v · v

v

=O(1)

m

X

v=1

|λv||sv|^k v

=O(1)

m−1

X

v=1

X_vβ_v +O(1)X_m|λ_m|=O(1) as m → ∞.

Therefore we get

m

X

n=1

P_n p_n

k−1

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

This completes the proof of Theorem3.1.

Finally if we takep_n = 1 for all values ofn in the theorem, then we obtain a new result concerning the|C,1|_ksummability factors.

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