ON QUASI β -POWER INCREASING SEQUENCES

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Quasiβ-Power Increasing Sequences Santosh Kr. Saxena vol. 10, iss. 2, art. 56, 2009

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ON QUASI β -POWER INCREASING SEQUENCES

SANTOSH Kr. SAXENA

H. N. 419, Jawaharpuri, Badaun Department of Mathematics Teerthanker Mahaveer University Moradabad, U.P., India

EMail:ssumath@yahoo.co.in

Received: 31 January, 2008

Accepted: 15 May, 2009

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 40D05, 40F05.

Key words: Absolute Summability, Summability Factors, Infinite Series.

Abstract: In this paper we prove a general theorem on

N , p¯ ^α_n;δ

k summability, which generalizes a theorem of Özarslan [6] on

N , p¯ n;δ

ksummability, under weaker conditions and by using quasiβ-power increasing sequences instead of almost increasing sequences.

Acknowledgements: The author wishes to express his sincerest thanks to Dr. Rajiv Sinha and the referees for their valuable suggestions for the improvement of this paper.

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1. Introduction

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

(1.1) Kn^βγ_n≥m^βγ_m

holds for alln≥m ≥1. It should be noted that every almost increasing sequence is a quasiβ-power increasing sequence for any non-negative β, but the converse need not be true as can be seen by taking the example, sayγ_n = n^−β forβ > 0. So we are weakening the hypotheses of the theorem of Özarslan [6], replacing an almost increasing sequence by a quasiβ-power increasing sequence.

Let P

a_n be a given infinite series with partial sums (s_n) and let(p_n) be a sequence withp₀ >0,p_n≥0forn >0andP_n=Pn

ν=0p_ν. We define (1.2) p^α_n =

n

X

ν=0

A^α−1_n−νp_ν, P_n^α =

n

X

ν=0

p^α_ν, P_−i^α =p^α_−i = 0, i≥1 ,

where

A^α₀ = 1, A^α_n = (α+ 1) (α+ 2)· · ·(α+n)

n! , (α >−1, n= 1,2,3, ...) (1.3)

The sequence-to-sequence transformation

(1.4) U_n^α = 1

P_n^α

n

X

ν=0

p^α_νs_ν

defines the sequence(U_n^α)of the N , p¯ ^α_n

mean of the sequence(s_n), generated by the sequence of coefficients(p^α_n)(see [7]).

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The seriesP

a_nis said to be summable N , p¯ ^α_n

k,k ≥1, if (see [2]) (1.5)

∞

X

n=1

P_n^α p^α_n

k−1

U_n^α−U_n−1^α

k <∞,

and it is said to be summable

N , p¯ ^α_n;δ

k,k ≥1andδ ≥0,if (see [7]) (1.6)

∞

X

n=1

P_n^α p^α_n

δk+k−1

U_n^α−U_n−1^α

k <∞.

In the special case when δ = 0, α = 0 (respectively, p_n = 1 for all values of n)

N , p¯ ^α_n;δ

ksummability is the same as N , p¯ _n

k(respectively|C,1;δ|_k) summability.

Mishra and Srivastava [4] proved the following theorem for|C,1|_ksummability.

Later on Bor [3] generalized the theorem of Mishra and Srivastava [4] for N , p¯ _n

k

summability.

Quite recently Özarslan [6] has generalized the theorem of Bor [3] under weaker conditions. For this, Özarslan [6] used the concept of almost increasing sequences.

A positive sequence (bn) is said to be almost increasing if there exists a positive increasing sequence(cn)and two positive constantsAandBsuch thatAcn ≤bn ≤ Bc_n(see [1]). Obviously every increasing sequence is an almost increasing sequence but the converse needs not be true as can be seen from the exampleb_n=ne⁽⁻¹⁾ⁿ. Theorem 1.1. Let(X_n)be an almost increasing sequence and the sequences (ρ_n) and(λ_n)such that the conditions

|∆λ_n| ≤ρ_n, (1.7)

ρ_n→0 as n→ ∞, (1.8)

|λ_n|X_n=O(1), as n→ ∞, (1.9)

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∞

X

n=1

n|∆ρ_n|X_n<∞.

(1.10)

are satisfied. If(p_n)is a sequence such that the condition

(1.11) P_n=O(np_n), as n→ ∞,

is satisfied and

(1.12)

m

X

n=1

P_n p_n

δk−1

|sn|^k=O(Xm), as m→ ∞,

(1.13)

∞

X

n=ν+1

P_n p_n

^δk−1 1

P_n−1 =O (

P_ν p_ν

δk

1 P_ν

) ,

then the seriesP

anλnis summable

N , p¯ _n;δ

kfork ≥1and0≤δ < ¹_k.

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2. Main Result

The aim of this paper is to generalize Theorem1.1for

N , p¯ ^α_n;δ

ksummability under weaker conditions by using quasi β-power increasing sequences instead of almost increasing sequences. Now, we will prove the following theorem.

Theorem 2.1. Let(Xn)be a quasiβ-power increasing sequence for some0< β <1 and the sequences(ρ_n)and(λ_n)such that the conditions (1.7) – (1.10) of Theorem 1.1are satisfied. If p^α_n

is a sequence such that

(2.1) P_n^α=O(np^α_n), as n→ ∞, is satisfied and

(2.2)

m

X

n=1

P_n^α p^α_n

δk−1

|sn|^k=O(Xm), as m→ ∞,

(2.3)

∞

X

n=ν+1

P_n^α p^α_n

δk−1

1

P_n−1^α =O (

P_ν^α p^α_ν

δk

1 P_ν^α

) , then the seriesP

a_nλ_nis summable

N , p¯ ^α_n;δ

_kfork≥1and0≤δ < _k¹.

Remark 1. It may be noted that, if we take(X_n)as an almost increasing sequence and α = 0 in Theorem2.1, then we get Theorem1.1. In this case, conditions (2.1) and (2.2) reduce to conditions (1.11) and (1.12) respectively and condition (2.3) reduces to (1.13). If additionallyδ = 0, relation (2.3) reduces to

(2.4)

∞

X

n=ν+1

p_n P_nPn−1

=O 1

P_ν

, which always holds.

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

Lemma 2.2 ([5]). Under the conditions on (X_n), (β_n) and (λ_n) as taken in the statement of the theorem, the following conditions hold

(2.5) nρnXn=O(1), as n→ ∞,

(2.6)

∞

X

n=1

ρ_nX_n <∞.

Proof of Theorem2.1. Let(T_n^α)be the N , p¯ ^α_n

mean of the seriesP

a_nλ_n.Then by definition, we have

T_n^α= 1 P_n^α

n

X

ν=0

p^α_ν

ν

X

w=0

a_wλ_w = 1 P_n^α

n

X

ν=0

P_n^α−P_ν−1^α a_νλ_ν.

Then, forn≥1, we get

T_n^α−T_n−1^α = p^α_n P_n^αP_n−1^α

n

X

ν=1

P_ν−1^α aνλν.

Applying Abel’s transformation, we have T_n^α−T_n−1^α = p^α_n

P_n^αP_n−1^α

n−1

X

ν=1

∆ P_ν−1^α λν

sν + p^α_n P_n^αsnλn

=− p^α_n P_n^αP_n−1^α

n−1

X

ν=1

p^α_νs_νλ_ν + p^α_n P_n^αP_n−1^α

n−1

X

ν=1

P_ν^αs_ν∆λ_ν + p^α_n P_n^αs_nλ_n

=T_n,1^α +T_n,2^α +T_n,3^α , say.

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Since

T_n,1^α +T_n,2^α +T_n,3^α

k≤3^k T_n,1^α

k+ T_n,2^α

k+ T_n,3^α

k , to complete the proof of Theorem2.1, it is sufficient to show that

∞

X

n=1

P_n^α p^α_n

δk+k−1

T_n,w^α

k <∞, for w= 1,2,3.

Now, whenk >1,applying Hölder’s inequality with indiceskandk⁰, where_k¹+_k¹0 = 1, and using|λ_n|=O

1 Xn

=O(1), by (1.9), we have

m+1

X

n=2

P_n^α p^α_n

δk+k−1

T_n,1^α

k =

m+1

X

n=2

P_n^α p^α_n

δk+k−1

p^α_n P_n^αP_n−1^α

n−1

X

ν=1

p^α_νs_νλ_ν

k

≤

m+1

X

n=2

P_n^α p^α_n

δk−1

1 P_n−1^α

n−1

X

ν=1

p^α_ν |s_ν|^k|λ_ν|^k 1 P_n−1^α

n−1

X

ν=1

p^α_ν

!^k−1

=O(1)

m

X

ν=1

p^α_ν|s_ν|^k|λ_ν|^k

m+1

X

n=ν+1

P_n^α p^α_n

^δk−1 1 P_n−1^α

=O(1)

m

X

ν=1

p^α_ν|s_ν|^k|λ_ν|^k P_ν^α

p^α_ν δk

1 P_ν^α

=O(1)

m

X

ν=1

P_ν^α p^α_ν

δk−1

|s_ν|^k|λ_ν|^k

=O(1)

m

X

ν=1

P_ν^α p^α_ν

δk−1

|s_ν|^k|λ_ν| |λ_ν|^k−1

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=O(1)

m

X

ν=1

P_ν^α p^α_ν

δk−1

|sν|^k|λν|

=O(1)

m−1

X

ν=1

∆|λ_ν|

ν

X

u=1

P_u^α p^α_u

δk−1

|s_u|^k+ O(1)|λ_m|

m

X

ν=1

P_ν^α p^α_ν

δk−1

|s_ν|^k

=O(1)

m−1

X

ν=1

|∆λ_ν|X_ν +O(1)|λ_m|X_m

=O(1)

m−1

X

ν=1

ρνXν +O(1)|λ_m|Xm

=O(1), as m→ ∞,

by virtue of the hypotheses of Theorem2.1and Lemma2.2. Sinceνρ_ν =O

1 Xν

= O(1),by (2.5), using the fact that|∆λn| ≤ ρnby (1.7) andP_n^α =O(np^α_n)by (2.1) and after applying the Hölder’s inequality again, we obtain

m+1

X

n=2

P_n^α p^α_n

δk+k−1

T_n,2^α

k

≤

m+1

X

n=2

P_n^α p^α_n

^δk−1 1 P_n−1^α

k(_n−1 X

ν=1

P_ν^α|∆λ_ν| |s_ν| )k

≤

m+1

X

n=2

P_n^α p^α_n

^δk−1 1 P_n−1^α

(_n−1 X

ν=1

p^α_ν (νρ_ν)^k|s_ν|^k

) ( 1 P_n−1^α

n−1

X

ν=1

p^α_ν )^k−1

=O(1)

m

X

ν=1

p^α_ν (νρ_ν)^k|s_ν|^k

m+1

X

n=ν+1

P_n^α p^α_n

δk−1

1 P_n−1^α

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=O(1)

m

X

ν=1

P_ν^α p^α_ν

δk−1

(νρ_ν)^k|s_ν|^k

=O(1)

m−1

X

ν=1

∆ (νρ_ν)

ν

X

w=1

P_w^α p^α_w

δk−1

|s_w|^k+O(1)mρ_m

m

X

ν=1

P_ν^α p^α_ν

δk−1

|s_ν|^k

=O(1)

m−1

X

ν=1

|∆ (νρν)|Xν +O(1)mρmXm

=O(1)

m−1

X

ν=1

ν|∆ρ_ν|X_ν+O(1)

m−1

X

ν=1

ρ_ν+1X_ν+1+O(1)mρ_mX_m

=O(1), as m → ∞,

by the virtue of the hypotheses of Theorem 2.1and Lemma 2.2. Finally, using the fact thatP_n^α =O(np^α_n),by (2.1) as inT_n,1^α , we have

m

X

n=1

P_n^α p^α_n

δk+k−1

T_n,3^α

k =O(1)

m

X

n=1

P_n^α p^α_n

δk−1

|s_n|^k|λ_n|

=O(1), asm → ∞.

Therefore, we get

∞

X

n=1

P_n^α p^α_n

δk+k−1

T_n,w^α

k=O(1), as m → ∞, for w= 1,2,3.

This completes the proof of Theorem2.1.

If we takep_n = 1andα= 0for all values ofnin Theorem2.1, then we obtain a result concerning the|C,1, δ|_k summability.

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References

[1] S. ALJAN ˇCI ´C AND D. ARANDELOVI ´C, O-regular varying functions, Publ.

Inst. Math., 22(36) (1977), 5–22.

[2] H. BOR, A note on some absolute summability methods, J. Nigerian Math. Soc., 6 (1987), 41–46.

[3] H. BOR, A note on absolute summability factors, Internet J. Math. and Math.

Sci., 17(3) (1994), 479–482.

[4] K.N. MISHRA AND R.S.L. SRIVASTAVA, On absolute Cesàro summability factors of infinite series, Portugal. Math., 42(1) (1985), 53–61.

[5] L. LEINDLER, A new application of quasi power increaseing sequences, Publ.

Math. (Debrecen), 58 (2001), 791–796.

[6] H.S. ÖZARSLAN , On almost increasing sequences and its applications, Inter- nat. J. Math. and Math. Sci, 25(5) (2001), 293–298.

[7] S.K. SAXENAANDS.K. SAXENA, A note on

N , p¯ ^α_n;δ

ksummability factors, Soochow J. Math., 33(4) (2007), 829–834.