1. Introduction. Let P a n be a given infinite series with the sequence of partial sums (s n ). By σ δn we denote the nth Ces` aro mean of order δ > −1 of the sequence (s n ),

VOL. 80 1999 NO. 2

A GENERAL THEOREM COVERING MANY ABSOLUTE SUMMABILITY METHODS

BY

W. T. S U L A I M A N (QATAR)

Abstract. A general theorem concerning many absolute summability methods is proved.

1. Introduction. Let P a _n be a given infinite series with the sequence of partial sums (s n ). By σ ^δ _n we denote the nth Ces` aro mean of order δ > −1 of the sequence (s n ),

σ ^δ _n = 1 A ^δ _n

n

X

v=1

A ^δ−1 _n−v s v . Here A ^δ _k = ^k+δ _k

= (δ + 1) . . . (δ + k)/k!. It may be easily verified that A ^δ _k ∼ k ^δ . The series P a n is said to be |C, δ| k summable, k ≥ 1, if

∞

X

n=1

n ^k−1 |σ _n ^δ − σ _n−1 ^δ | ^k < ∞.

Let (p n ) be a sequence of positive numbers such that P n =

n

X

v=0

p v → ∞ as n → ∞.

The transformation

t n = 1 P n

∞

X

v=0

p v s v

defines the sequence (t n ) of the Riesz means of the sequence (s n ) generated by the sequence of coefficients (p n ) (see [4]). The series P a n is said to be

|R, p _n | _k summable, k ≥ 1, if

∞

X

n=1

n ^k−1 |t _n − t _n−1 | ^k < ∞.

1991 Mathematics Subject Classification: 40D15, 40F15, 40F05.

Key words and phrases: summability, series, sequence.

[245]

The series P a _n is said to be |N , p n | _k summable, k ≥ 1 (Bor [1]), if

∞

X

n=1

 P n

p n

 k−1

|t n − t n−1 | ^k < ∞.

In the special case when p n = 1 for all values of n, both |R, p n | _k and

|N , p n | k summability are the same as |C, 1| k summability.

The series P a _n is said to be |N, p n | summable if (1)

∞

X

n=1

|T _n − T _n−1 | < ∞, where

T n = 1 P n

n

X

v=0

p n−v s v . The sequence class M is defined by

M =



p = {p n } : p _n > 0 & p n+1

p n

≤ p n+2

p n+1

≤ 1, n = 0, 1, . . . , P _n → ∞

 . It is known (Das [3]) that for p ∈ M , (1) holds iff

∞

X

n=1

1 nP n

  

n

X

v=1

p n−v va v

   < ∞.

For p ∈ M , the series P a n is said to be |N, p n | _k summable, k ≥ 1 (Sulaiman [5]), if

∞

X

n=1

1 nP _n ^k

  

n

X

v=1

p n−v va v

  

k

< ∞.

In the special case in which p n = A ^r−1 _n , r > −1, |N, p n | _k summability is equivalent to |C, r| k summability. The series P a n is said to be |R, log n, 1| k

summable if it is |N , p n | _k summable with p n = 1/(n+1) and P n ∼ log(n+1).

For any sequence {f n }, we define ∆f _n = f n − f _n+1 . 2. Main result. We prove the following:

Theorem 1. Let {f n }, {g _n }, {G _n }, and {H _n } be sequences of positive constants such that {f n } ∈ M and F n = P n

v=1 f v → ∞ as n → ∞. Let {ε _n } be a sequence of constants. Given a sequence {x _n } define

X n = 1 G n

n

X

v=1

g v x v , Y n = 1 F n−1 H n

n

X

v=1

vf n−v x v ε n

and assume

g n+1 = O(g n ), (2)

H n+1 F n+1

G n+1

= O  H _n F n

G n



,

(3)

∆g n = O  g n+1

n

 , (4)

∆  g n H n F n

nG n



= O  g n H n

nG n

 , (5)

∆

 nG n

g n H n F n

ε n



= O

 1 F n

 , (6)

∞

X

n=v+1

f n−v

F n−1 H _n ^k = O

 1 H _v ^k

 . (7)

Let k ≥ 1. Then a necessary and sufficient condition for the implication:

if X

|X n | ^k < ∞ then X

|Y n | ^k < ∞ to hold (for any sequence {x n }) is

(i) ε n = O(g n H n F n /(nG n )), and (ii) ∆ε n = O(g n+1 H n /(nG n )).

3. Lemmas

Lemma 1 (Bor [2]). Let k ≥ 1, and let A = (a nv ) be an infinite matrix that maps ` <sup>k</sup> into ` ^k . Then a nv = O(1) for all n and v.

P r o o f. By the Closed Graph Theorem, A defines a bounded linear map- ping in ` ^k . Then the bound |a nv | ≤ C follows, where C is the norm of A.

Lemma 2 (Sulaiman [6]). Let p ∈ M. Then for 0 < r ≤ 1,

∞

X

n=v+1

p n−v−1

n ^r P n−1

= O(v ^−r ).

Lemma 3. Suppose that ε ⁿ = O(α n β n ), α n , β n > 0, α n+1 β n+1 = O(α n β n ), ∆(α n β n ) = O(α n ) and ∆(ε n /(α n β n )) = O(1/β n ). Then ∆ε n = O(α n ).

P r o o f. We have ε n = k n α n β n where k n = ε n /(α n β n ) = O(1). Therefore

∆ε n = k n ∆(α n β n ) + ∆k n (α n+1 β n+1 )

= O(1)O(α n ) + O(1/β n )O(α n β n ) = O(α n ).

4. Proof of Theorem 1. Sufficiency. We have via Abel’s transforma- tion:

Y n = 1 F n−1 H n

n

X

v=1

g v x v

 v f n−v

g v

ε v



= 1

F n−1 H n

 ⁿ⁻¹ X

v=1

 X ^v

r=1

g r x r



∆ v

 v f n−v

g v

ε v

 +

 X ⁿ

r=1

g r x r

 n f 0

g n

ε n



= 1 F n−1 H n

n−1

X

v=1

G v X v



− f n−v

g v

ε v + (v + 1)∆g ⁻¹ _v f n−v ε v

+ (v + 1)g _v+1 ⁻¹ ∆ v f n−v ε v + (v + 1)g _v+1 ⁻¹ f n−v−1 ∆ε v



+ nG n X n f 0

F n−1 H n g n

ε n

= Y n,1 + Y n,2 + Y n,3 + Y n,4 + Y n,5 , say.

By Minkowski’s inequality,

m

X

n=1

|Y n,1 | ^k = O(1)

m

X

n=1 5

X

r=1

|Y n,r | ^k . Applying H¨ older’s inequality gives

m+1

X

n=2

|Y n,1 | ^k =

m+1

X

n=2

1 F _n−1 ^k H _n ^k

   

n−1

X

v=1

f n−v

G v

g v

X v ε v

   

k

≤

m+1

X

n=2

1 F n−1 H _n ^k

×

 ⁿ⁻¹ X

v=1

f n−v

 G v

g v

 k

|X _v | ^k |ε _v | ^k

 ⁿ⁻¹ X

v=1

f n−v

F n−1

 k−1

≤ O(1)

m

X

v=1

 G v

g v

 k

|X _v | ^k |ε _v | ^k

m+1

X

n=v+1

f n−v

F n−1 H _n ^k

≤ O(1)

m

X

v=1

1 H _v ^k

 v F v

 k

 G v

g v

 k

|X _v | ^k |ε _v | ^k ,

m+1

X

n=2

|Y _n,2 | ^k =

m+1

X

n=2

1 F _n−1 ^k H _n ^k

  

n−1

X

v=1

(v + 1)G v ∆g _v ⁻¹ f n−v X v ε v

  

k

≤ O(1)

m+1

X

n=2

1 F n−1 H _n ^k

× n ⁿ⁻¹ X

v=1

v ^k G ^k _v |∆g ⁻¹ _v | ^k f n−v |X v | ^k |ε v | ^k o  n−1

X

v=1

f n−v

F n−1

 k−1

≤ O(1)

m

X

v=1

v ^k G ^k _v |∆g ⁻¹ _v ||X _v | ^k |ε _v | ^k

m+1

X

n=v+1

f n−v

F n−1 H _n ^k

= O(1)

m

X

v=1

 v H v

 k

G ^k _v |∆g v | ^k

g _v ^k g ^k _v+1 |X v | ^k |ε v | ^k

≤ O(1)

m

X

v=1

1 H _v ^k

 v F v

 k

 G v

g v

 k

|X _v | ^k |ε _v | ^k ,

m+1

X

n=2

|Y n,3 | ^k =

m+1

X

n=2

1 F _n−1 ^k H _n ^k

  

n−1

X

v=1

(v + 1)g _v+1 ⁻¹ G v ∆ v f n−v X v ε v

  

k

≤

m+1

X

n=2

1 F _n−1 ^k H _n ^k

×

 ⁿ⁻¹ X

v=1

v ^k

 G v

g v+1

 k

|∆ _v f n−v ||X _v | ^k |ε _v | ^k

 n ⁿ⁻¹ X

v=1

|∆f _n−v | o k−1

≤ O(1)

m

X

v=1

v ^k  G v

g v

 k

|X _v | ^k |ε _v | ^k

m+1

X

n=v+1

|∆ _v f n−v | F _n−1 ^k H _n ^k

≤ O(1)

m

X

v=1

1 H _v ^k

 v F v

 k

 G v

g v

 k

|X _v | ^k |ε _v | ^k ,

m+1

X

n=2

|Y _n,4 | ^k =

m+1

X

n=2

1 F _n−1 ^k H _n ^k

  

n−1

X

v=1

vg _v+1 ⁻¹ f n−v−1 G v X v ∆ε v

  

k

≤

m+1

X

n=2

1 F n−1 H _n ^k

×

 n−1

X

v=1

v ^k

 G v

g v+1

 k

f n−v−1 |X v | ^k |∆ε v | ^k

 n−1

X

v=1

f n−v−1

F n−1

 k−1

≤ O(1)

m

X

v=1

v ^k

 G v

g v+1

 k

|X _v | ^k |∆ε _v | ^k

m+1

X

n=v+1

f n−v−1

F n−1 H _n ^k

≤ O(1)

m

X

v=1

 v H v

 k  G v

g v+1

 k

|X v | ^k |∆ε v | ^k ,

m

X

n=1

|Y _n,5 | ^k =

m

X

n=1

   

nG n X n f 0 ε n

F n−1 H n g n

   

k

≤ O(1)

m

X

n=1

 n F n

 k

 G n

g n

 k

1

H _n ^k |X _n | ^k |ε _n | ^k .

Necessity of (i). By the result of Bor [1], the transformation from (X n )

into (Y n ) maps ` <sup>k</sup> into ` ^k and hence the diagonal elements of this transfor-

mation are bounded (by Lemma 1), so (i) is necessary.

Necessity of (ii). This follows from Lemma 3 and necessity of (i) by taking α n ≡ g _n H n /(nG n ) and β n ≡ F _n , using (2).

This completes the proof of the theorem.

Remark. (1) If we put x ⁿ = a n , f n = p n and H n = n ^1/k in the formula defining Y n , p ∈ M , then the condition P |Y n | ^k < ∞ is equivalent to |N, p n | _k summability of P a _n ε n (note that P n /P n−1 is a bounded sequence).

(2) If we put x n = a n , Q n = q 0 + . . . + q n , g n = Q n−1 and G n = Q n−1 (Q n /q n ) ^1/k in the formula defining X n , then the condition P |X n | ^k <

∞ simply means |N , q _n | _k summability of P a _n .

(3) If we put x n = a n , Q n = q 0 + . . . + q n , g n = Q n−1 and G n = n ^1/k−1 Q n Q n−1 /q n in the formula defining X n , then the condition P |X _n | ^k

< ∞ means |R, q n | _k summability of P a _n .

5. Applications. Throughout the rest of the paper we assume that P n → ∞ and Q n → ∞ as n → ∞.

Theorem 2. Let p ∈ M and let nq n = O(Q n ), Q n = O(Q n−1 ), and P n+1

P n

= O

 Q n

Q n−1

 q n Q n+1

q n+1 Q n

 1/k  ,

∆  P n

n

 nq n

Q n

 1/k 

= O  1 n

 nq n

Q n

 1/k  ,

∆  n P n

 Q n

nq n

 1/k

ε n



= O

 1 P n

 .

Then a necessary and sufficient condition that P a _n ε n be |N, p n | _k summable whenever P a _n is |N , q n | _k summable, k ≥ 1, is

ε n = O  P n

n

 nq n

Q n

 1/k 

, ∆ε n = O  1 n

 nq n

Q n

 1/k  . Theorem 3. Let p ∈ M and let nq n = O(Q n ), Q n = O(Q n−1 ), and

P n+1

P n

= O  q n Q n+1

q n+1 Q n



, ∆  P n q n

Q n



= O  q n

Q n

 ,

∆

 Q n

P n q n

ε n



= O

 1 P n

 .

Then a necessary and sufficient condition that P a _n ε n be |N, p n | _k summable whenever P a n is |R, q n | _k summable, k ≥ 1, is

ε n = O(P n q n /Q n ), ∆ε n = O(q n /Q n ).

The following results are consequences of Theorem 2.

Corollary 4. A necessary and sufficient condition that P a _n ε n be

|C, α| _k summable, 0 < α < 1, whenever P a _n is |C, 1| k summable, k ≥ 1, is ε n = O(n ^α−1 ), ∆ε n = O(n ⁻¹ ),

provided that ∆(n ^1−α ε n ) = O(n ^−α ).

Corollary 5. A necessary and sufficient condition that P a _n ε n be

|N, 1/(n + 1)| _k summable whenever P a n is |C, 1| k summable, k ≥ 1, is ε n = O(log n/n), ∆ε n = O(n ⁻¹ ),

provided that

∆

 n log n ε n



= O

 1 log n

 .

Corollary 6. A necessary and sufficient condition that P a n ε n be

|N, 1/(n + 1)| _k summable whenever P a n is |R, log n, 1| k summable, k ≥ 1, is

ε n = O{(log n) ^1−1/k /n}, ∆ε n = O{1/n(log n) ^1/k }, provided that

∆

 n

(log n) ^1−1/k ε n



= O

 1 log n

 .

Corollary 7. A necessary and sufficient condition that P a _n ε n be

|C, α| _k summable, 0 < α ≤ 1, whenever P a n is |R, log n, 1| k summable, k ≥ 1, is

ε n = O{n ^α−1 /(log n) ^1/k }, ∆ε n = O{1/(n(log n) ^1/k )}, provided that ∆(n ^1−α (log n) ^1/k ε n ) = O(n ^−α ).

Acknowledgments. The author is grateful to the referee for his kind advice and valuable suggestions during the preparation of this paper.

REFERENCES

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n

|

_k

summability factors, Proc. Amer. Math. Soc. 94 (1985), 419–

422.

[2] —, On the relative strength of two absolute summability methods, ibid. 113 (1991), 1009–1012.

[3] G. D a s, Tauberian theorems for absolute N¨ orlund summability , Proc. London Math.

Soc. 19 (1969), 357–384.

[4] G. H. H a r d y, Divergent Series, Oxford Univ. Press, Oxford, 1949.

[5] W. T. S u l a i m a n, Notes on two summability methods, Pure Appl. Math. Sci. 31

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Department of Mathematics College of Science

University of Qatar Qatar

E-mail: waad@qu.edu.qa

Received 23 January 1998;

revised 2 December 1998