Also we have obtained a new result concerning the|C,1|ksummability factors

A GENERAL NOTE ON INCREASING SEQUENCES

HÜSEY˙IN BOR

DEPARTMENT OFMATHEMATICS

ERCIYESUNIVERSITY

38039 KAYSERI, TURKEY bor@erciyes.edu.tr

URL:http://fef.erciyes.edu.tr/math/hbor.htm

Received 23 December, 2006; accepted 08 August, 2007 Communicated by S.S. Dragomir

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

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

2000 Mathematics Subject Classification. 40D15, 40F05, 40G99.

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 constantsA and B such that Ac_n ≤ b_n ≤ Bc_n (see [1]). We denote byBVOthe expressionBV ∩ CO, whereCOandBV are the set of all null sequences and the set of all sequences with bounded variation, respectively. LetP

a_nbe a given infinite series with partial sums(s_n). We denote byu^α_nandt^α_nthen-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 <∞.

009-07

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).

The sequence-to-sequence transformation

(1.6) σ_n= 1

Pn n

X

v=0

p_vs_v

defines the sequence(σ_n)of the Riesz mean or simply the( ¯N , p_n)mean of the sequence(s_n), generated by the sequence of coefficients(p_n)(see [7]). The seriesP

a_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|∆σ_n−1|^k <∞,

where

(1.8) ∆σn−1 =− p_n

P_nP_n−1

n

X

v=1

Pv−1a_v, n≥1.

In the special case pn = 1 for all values of n, N , p¯ n

k summability is the same as |C,1|_k summability.

2. KNOWN RESULTS

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

summa- bility factors.

Theorem A. Let(X_n)be a positive non-decreasing sequence and let there be sequences(β_n) and(λ_n)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=1a_n^P_np^nλn

n is summable N , p¯ _n

. Later on Bor [4] generalized Theorem A for

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 Theorem A are 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_np^nλn

n is summable N , p¯ _n

k, k≥1.

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

Quite recently Bor [5] has proved Theorem B under 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 seriesP∞

n=1a_n^P_np^nλn

n is summable N , p¯ _n

_k,k ≥1.

Remark 2.1. It should be noted that, under the conditions of Theorem B,(λ_n)is bounded and

∆λn=O(1/n)(see [4]).

3. MAINRESULT

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(γn)is said to be a quasi β-power increasing sequence if there exists a constantK =K(β, γ)≥1such that

(3.1) Kn^βγ_n≥m^βγ_m

holds for all n ≥ m ≥ 1. It should be noted that almost every increasing sequence is a quasi β-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 quasi β-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=1a_n^P_np^nλn

n is summable N , p¯ _n

k,k ≥1.

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

We require the following lemma for the proof of Theorem 3.1.

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) nX_nβ_n =O(1),

(3.4)

∞

X

n=1

β_nX_n<∞.

Proof of Theorem 3.1. Let(T_n)denote the( ¯N , p_n)mean of the series P∞ 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−P_v−1)a_vP_vλ_v vp_v ,

and thus

(3.6) T_n−Tn−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−Tn−1 = p_n P_nP_n−1

n

X

v=1

s_v∆

Pv−1P_vλ_v vp_v

+λ_ns_n n

= s_nλ_n

n + p_n P_nP_n−1

n−1

X

v=1

s_vP_v+1P_v∆λ_v (v+ 1)p_v+1

+ p_n PnPn−1

n−1

X

v=1

P_vs_vλ_v∆ P_v

vpv

− p_n PnPn−1

n−1

X

v=1

s_vP_vλ_v1 v

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

To prove Theorem 3.1, by Minkowski’s inequality, it is sufficient to show that

(3.7)

∞

X

n=1

P_n p_n

k−1

|T_n,r|^k <∞, for r = 1,2,3,4.

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

βnXn+O(1)|λm|Xm =O(1) as m → ∞,

by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2.

Now, using the fact that P_v+1 = O((v + 1)p_v+1), by (2.6), and then applying Hölder’s in- equality, 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 PnP_n−1^k

n−1

X

v=1

P_vs_v∆λ_v

k

=O(1)

m+1

X

n=2

pn

P_nP_n−1^k (_n−1

X

v=1

Pv

p_v |sv|pv|∆λv| )k

=O(1)

m+1

X

n=2

p_n PnPn−1

n−1

X

v=1

P_v pv

k

|s_v|^kp_v|∆λ_v|^k 1 Pn−1

n−1

X

v=1

p_v

!k−1

=O(1)

m

X

v=1

P_v pv

k

|s_v|^kp_v|∆λ_v|^k

m+1

X

n=v+1

p_n PnPn−1

=O(1)

m

X

v=1

P_v|∆λ_v| pv

^k−1

|s_v|^k|∆λ_v|

=O(1)

m

X

v=1

|s_v|^k|∆λ_v| 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

|∆(vβv)|Xv+O(1)mβmXm

=O(1)

m−1

X

v=1

v|∆βv|Xv+O(1)

m−1

X

v=1

|βv|Xv+O(1)mβmXm =O(1) asm → ∞, in view of the hypotheses of Theorem 3.1 and Lemma 3.2.

Again, since∆(_vp^Pv

v) =O(¹_v),by (2.6) and (2.7) (see [10]), as inTn,1 we 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 PnP_n−1^k

(_n−1 X

v=1

P_v|s_v| |λ_v|1 v

)k

=O(1)

m+1

X

n=2

pn

P_nP_n−1^k (_n−1

X

v=1

Pv

p_v

p_v|s_v| |λ_v|1 v

)k

=O(1)

m+1

X

n=2

p_n PnPn−1

n−1

X

v=1

P_v vpv

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_nPn−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 vp_v

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

Pn

p_n k−1

|T_n,4 |^k=

m+1

X

n=2

pn

P_nP_n−1^k

n−1

X

v=1

s_vPv

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 vp_vp_vλ_v

k

≤

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

Pv

vp_v k

|s_v|^kp_v|λ_v|^k 1 P_v · v

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 → ∞.

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 Theorem 3.1.

Finally if we take pn = 1 for all values of n in the theorem, then we obtain a new result

concerning the|C,1|_ksummability factors.

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