Inclusion Theorems for Absolute Matrix Summability Methods

Journal of Mathematical Analysis and Applications 238, 82᎐90 Ž1999. Article ID jmaa.1999.6507, available online at http:rrwww.idealibrary.com on

Inclusion Theorems for Absolute Matrix Summability Methods B. E. Rhoades Department of Mathematics, Indiana Uni¨ ersity, Bloomington, Indiana 47405-7106 Submitted by William F. Ames Received February 9, 1999

In this paper we establish a general inclusion theorem for a nonnegative lower triangular matrix to be absolutely stronger than a weighted mean matrix. Several inclusion theorems for Cesaro ` and weighted mean methods are then obtained as corollaries. 䊚 1999 Academic Press Key Words: absolute summability matrix; Cesaro ` matrix; weighted mean matrix.

Let ⌺ a n be a given series with partial sums  sn4 , Ž C, ␣ . the Cesaro ` matrix of order ␣ . If ␴n␣ denotes the nth term of the Ž C, ␣ .-transform of  sn4 , then, from Flett w5x, Ýa n is said to be summable < C, ␣ < k , k G 1 if ⬁

␣ < ky 1 - ⬁. Ý n ky1 < ␴n␣ y ␴ny1

Ž 1.

ns1

For any sequence  u n4 , the forward difference operator ⌬ is defined by ⌬ u n s u n y u nq1. An appropriate extension of Ž1. to arbitrary lower triangular matrices would be ⬁

Ý n ky1 < ⌬ t ny 1 < ky 1 - ⬁,

Ž 2.

ns1

where n

tn [

Ý a n k sk . ks0

Such an extension is used in w3x. However, in w4x, Bor and Thorpe make the following extension of Ž1.. A series ⌺ a n is said to be summable 82 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

83

INCLUSION THEOREMS

< N, pn < k , k G 1 if ⬁

ky 1

Pn

< ⌬ Zny 1 < k - ⬁,

ž /

Ý

pn

ns1

Ž 3.

where N denotes the weighted mean method generated by the sequence  pn4 , with partial sums Pn , and Zn s Ž1rPn .Ý nks0 pk sk . It is clear that Ž3. cannot be an appropriate extension of Ž1., since it does not reduce to Ž1. when the matrix method is Ž C, ␣ .. In spite of this fact, several comparison theorems have been established using Ž3.. See, for examples w1, 2, 4, 6x. The purposes of this paper are to establish Theorem 1 of w6x using definition Ž2. instead of Ž3., and then show that certain results of Bor, Sarigol, ¨ and Thorpe can be obtained as corollaries. Given a lower triangular matrix T one defines T s Ž t ni . and Tˆ s Žˆt ni . by n

t ni s

Ý ar i

and

ˆt ni s t ni y t ny1, i ,

rsi

respectively. THEOREM.

Let T be a nonnegati¨ e lower triangular matrix satisfying

Ž i.

t ni G t nq1, i , n G i , i s 0, 1, . . . ,

Ž ii .

Pn t n n s O Ž pn . ,

Ž iii .

t n0 s t ny1, 0 , n s 1, 2, . . . , ny1

Ž iv.

Ý is1

pi

< t n , iq1 < s O Ž t n n . ,

ž /ˆ Pi

⬁

Ž v.

Ž nt n n .

ky1

Ý Ž ntn n .

ky1

Ý

< ⌬ t ny1 , i < s O

nsiq1

i ky 1 pik

ž / Pik

,

and ⬁

Ž vi.

nsiq1

< ˆt n , iq1 < s O

ipi

ž / Pi

ky 1

.

Then, if Ýa n is < N, p < k summable, it is < T < k summable, k G 1.

84

B. E. RHOADES

Proof. Condition Žiii. implies that the row sums of T are all the same. Without loss of generality we may assume that this number is one. Thus, Zn [

1

n

Pn

␯s0

Ý

n

Pn

␯ s0

Ý

p␯

Ý

ai s

is0

is0

is0

ny1

Ý ai is0

ž

1y

Piy1 Pny 1

/

.

/

q

ž

␯

1 Pn

n

Ý ai Ž Pn y Piy1 . s Ý ai

Pn

Zny 1 s

␯

1

n

1

s

p␯ s␯ s

1y

n

ai

Ý Ý p␯ is0

Piy1 Pn

/

␯ si

,

Therefore y⌬ Zny 1 s a n 1 y

ž

s s Pny 1 Pn ⌬ Zny1 pn

Pny 1 Pn

ny1 is0

ny1

a n pn

q

Pn

Ý ai is0

pn

ny1

Pny 1 Pn

is1

Ý ai

ž

1

ž

1y

y

Pny1

1 Pn

/

Pny1 Pn

y1q

Pny1 Pny1

/

Piy1

n G 1,

Ý ai Piy1 ,

n

s y Ý a i Piy1 , is1

and Pny 2 Pny1 ⌬ Zny2 pny 1

ny1

sy

Ý ai Piy1 . is1

Thus Pny 1 Pn ⌬ Zny1 pn

y

Pny2 Pny1 ⌬ Zny2 pny1

s ya n Pny1 ,

so that an s

Pny 2 ⌬ Zny2 pny 1

y

Pn ⌬ Zny1 pn

,

n G 1,

Ž 4.

n

⌬Tny 1 s y Ý a iˆt ni , is1

since t ny 1, n s 0.

Ž 5.

85

INCLUSION THEOREMS

Substituting Ž4. into Ž5. yields Pi ⌬ Ziy1

n

⌬Tny 1 s

Ý ˆt ni

pi

is1 n

s

Ý ˆt ni

Pi ⌬ Ziy1 pi

is1

ny1

Ý is1

1 pi

ny1

y

Ý ˆt n , rq1

y ˆt n1

pn

y

piy1 Pry1 ⌬ Zry1 pr

rs0

Pn ⌬ Zny1

s ˆt n n

Piy2 ⌬ Ziy2

y

Py1 ⌬ zy1 p0

Piˆt ni y Piy1ˆt n , iq1 ⌬ Ziy1 .

Note that ˆt n n s t n n y t ny1, n s t n n s t n n . Thus ⌬Tny 1 s ˆt n n

Pn ⌬ Zny1 pn ny1

y

Ý is1

1 pi

ny1

q

Ý is1

Pi

ž /Ž pi

ˆt ni y ˆt n , iq1 . ⌬ Ziy1

Ž Pi y Piy1 . ˆt n , iq1 ⌬ Ziy1

s Tn Ž 1 . q Tn Ž 2 . q Tn Ž 3 . , say Using Žii. and Ž2., ⬁

Ýn

ky1

Tn Ž 1 .

ns1

k

s

⬁

Ýn

ky1

t n n Pn pn

ns1

s O Ž 1.

k

⌬ Zny1

⬁

Ý n ky1 < ⌬ Zny 1 < k - ⬁. ns1

Using the definition of ˆt ni and Holder’s inequality, ¨ J2 [

⬁

Ý n ky1 Tn Ž 2.

k

ns2

F

⬁

Ý

ny1

n ky1

ns2

F

⬁

Ý is1

ny1

Ý Ý ns2 is1

Pi

ž / pi

Pi

ž / pi

k

< t ni y t ny1, i < < ⌬ Ziy1 <

k

< t ni y t ny1, i < < ⌬ Ziy1 < = n k

ny1

Ý < t ni y t ny1, i <

is1

ky 1

.

86

B. E. RHOADES

Using Ži. and Žiii., ny1

ny1

is1

is1

Ý < t ni y t ny1, i < s Ý Ž t ny1, i y t ni . s 1 y t ny1, 0 y Ž 1 y t n n y t n0 . s y Ž t ny 1, 0 y t n0 . q t n n F t n n .

Therefore ⬁

J2 F

Ý

ny1

ky1

Ž nt n n . k

Pi

Ý is1

pi

is1

⬁

s

⬁

< ⌬ Ziy1 < k

ž / pi

< t ni y t ny1, i < < ⌬ Ziy1 < k

ž /

Ý

ns2

k

Pi

Ý Ž ntn n . ky1 < t ni y t ny1, i < . nsiq1

Using Žv. and Ž2., ⬁

J2 s O Ž 1.

Ý is1

J3 [

k ky1

Pi

i

ž /

Pik

pi

⬁

Ýn

ky1

pik

k

Tn Ž 3 .

s

ns2

F

ny1

n ky1

ns2

Ý is1

Pi

⬁

Ý i ky1 < ⌬ Ziy1 < k - ⬁, is1

⬁

Ýn

Ý ˆt n , iq1 ⌬ Ziy1 is1

ky1

< ˆt n , iq1 < < ⌬ Ziy1 < k =

ž / pi

k

ny1 ky1

ns2

⬁

Ý

< ⌬ Ziy1 < k s O Ž 1 .

ny1

Ý is1

pi

ž / Pi

ky 1

< ˆt n , iq1 <

.

Using Živ., Ž1., and Ž2., J3 s O Ž 1.

⬁

Ý Ž ntn n .

ky1

ns2

s O Ž 1.

⬁

Ý is1

Pi

Pi

is1

pi

ky 1

ipi

nt n n s O Ž1.,

< ˆt n , iq1 < < ⌬ Ziy1 < k

pi

⬁

Ý Ž ntn n . ky1 <ˆt n , iq1 < nsiq1

ky1

Pi

COROLLARY 1 w7, Theorem 3.1x. Žii.

ky 1

ž /

< ⌬ Ziy1 < k

pi

⬁

Pi

ky 1

ž / Ýž / ž / Ý

is1

s O Ž 1.

ny1

< ⌬ Ziy1 < k s O Ž 1 .

⬁

Ý i ky1 < ⌬ Ziy1 < k - ⬁. is1

Let T satisfy Ži. and Žiii. of the theorem

87

INCLUSION THEOREMS

and there exists a constant C ) 0 such that

Živ. n

Ý Ž t ni y t ny1, i .

is ␯

F C < t n ␯ y t ny1 , ␯ <

for 0 F ␯ F n, n G 1.

Then, if Ýa n an is summable < C, 1 < k it is also summable < T < k , k G 1. Proof. Ž C, 1. is a weighted mean method with pn s 1 for all n, so that condition Žii. above implies condition Žii. of the theorem. Condition Živ. above implies condition Živ. of the theorem. It remains to show that conditions Žv. and Žvi. of the theorem are satisfied. Using Žii. and Ži., Pik

⬁

i ky1 pik

Ž nt n n .

Ý

ky1

⬁

< t ni y t ny1, i < s O Ž 1 . i

nsiq1

Ý Ž t ny1, i y t ni . nsiq1

F O Ž 1 . i w t ii y lim inf n t ni x F O Ž 1 . it ii s O Ž 1 . . For Žvi., Pi

ž / ipi

ky 1

⬁

Ý Ž ntn n .

ky1

< ˆt n , iq1 < s

nsiq1

ž

iq1

⬁

ky1

/

i

nsiq1

⬁

s O Ž 1.

O Ž 1 . < ˆt n , iq1 <

Ý i

Ý Ý Ž t ny 1, r y t n r . nsiq1 rs0 i

s O Ž 1.

⬁

Ý Ý Ž t ny 1, r y t n r . rs0 nsiq1 i

s O Ž 1.

Ý ti r s O Ž 1. . rs0

COROLLARY 2 w3, Theoremx. sequences such that

Ž i.

qn Pn pn Q n

Suppose that  pn4 and  qn4 are positi¨ e

s O Ž 1. ,

and ⬁

Ž ii .

Ý nsiq1

n ky 1qnk Q nk Q ny1

sO

ž

i ky1 qiky1 Q ik

/

.

Then, if Ýa n is summable < N, p < k it is summable < N, q < k , k G 1.

88

B. E. RHOADES

Proof. With T s Ž N, q ., conditions Ži. and Žiii. of the theorem are automatically satisfied. Condition Ži. of Corollary 2 implies condition Žii. of the theorem and condition Žv. of the theorem reduces to condition Žii. of the corollary. For condition Živ., 1 ny1 pi < ˆt <, J4 [ Ý t n n is1 Pi n , iq1

ž /

ˆt n , iq1 s t n , iq1 y t ny1, iq1 n

s

ny1

tn r y

Ý rsiq1

i

t ny1, r s 1 y

Ý rsiq1

rs0

i

s

i

Ý Ž t ny 1, i y t ni . s Ý rs0

s

rs0 i

qn

ž

qi

rs0

y

Q ny 1

qn Q i

Ý qr s Q

Q n Q ny1

i

Ý t r i y 1 q Ý t ny1, i

n Q ny1

rs0

qi Qn

/

.

Therefore J4 s

Qn

ny1

qn

is1

pi

qn Q i

Pi

Q n Q ny1

ž /

Ý

s

1

ny1

Q ny1

is1

Ý qi s O Ž 1 . .

For condition Žvi. of the theorem, using Žii. of the corollary, Pi

ky 1

Ý Ž ntn n . ky1 <ˆt n , iq1 <

ž / ipi

s s s

⬁

nsiq1

Pi

ky 1

⬁

nqn

ž / ž / ž / Ýž / ž / Ý ž / ž /

sO

sO

Ý

ipi Pi

Qn

⬁

nqn

nsiq1

Qn

ky 1

ipi Pi

nsiq1

ky 1

Pi

Pi qi

pi Q i

rs0 ky1

qn Q i Q n Q ny1

n ky1 qnk

nsiq1

Q nk Q ny1

ky 1

ipi

i

Ý Ž t ny 1, r y t n r .

⬁

Qi

ipi

ky1

Qi

i ky1 qiky1 Q ik

ky 1

s O Ž 1. .

89

INCLUSION THEOREMS

COROLLARY 3 w1x.

Let  pn4 be a positi¨ e sequence satisfying npn

Ž i.

Pn

s O Ž 1. ,

and Pn

Ž ii .

npn

s O Ž 1. .

Then, if Ýa n an is summable < C, 1 < k it is summable < N, p < k , k G 1. Conditions Ži. and Žii. of Corollary 3 imply that definitions Ž2. and Ž3. are equivalent. Consequently the result follows from w1x. However, for completeness, we show that Corollary 3 follows from Corollary 2. Proof. Using Corollary 2 with pn s 1 for all n, qn s pn , condition Ži. of Corollary 3 implies condition Ži. of Corollary 2. Using conditions Ži. and Žii. of Corollary 3, Q ik i ky1 qiky 1

⬁

Ý nsiq1

n ky 1qnk Q nk Q ny1

Pik

s

i ky1 pik

n ky 1 pnk

⬁

Ý nsiq1

Pnk Pny1

⬁

O Ž 1 . pn

s O Ž 1 . Pi

Ý nsiq1

Pny 1 Pn

⬁

1

s O Ž 1 . Pi

Ý nsiq1

ž

Pny 1

y

1 Pn

/

s O Ž 1. .

COROLLARY 4 w2, Theorem 1x. Let  pn4 be a positi¨ e sequence satisfying the conditions of Corollary 3. Then, if Ýa n is summable < N, p < k it is also summable < C, 1 < k , k G 1. Proof. Using qn s 1 for all n in Corollary 2, condition Ži. of Corollary 3 implies condition Ži. of Corollary 2. Hence, Q ik i ky1 qiky 1

⬁

Ý nsiq1

n ky1 qnk Q nk Q ny1

s

ž

Ž i q 1. i ky1

k

/

s O Ž 1. Ž i q 1.

⬁

Ý nsiq1 ⬁

n ky1 k Ž n q 1. n

O Ž 1.

s O Ž 1. . Ý nsiq1 Ž n q 1 . n

Combining Corollaries 3 and 4 one obtains the fact that, if  pn4 is a nonnegative sequence satisfying conditions Ži. and Žii. of Corollary 3, then summability < N, p < k and < C, 1 < k are equivalent for k G 1. This fact is Theorem 2 of w2x.

90

B. E. RHOADES

REFERENCES 1. H. Bor, On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 Ž1985., 147᎐149. 2. H. Bor, A note on two summability methods, Proc. Amer. Math. Soc. 98 Ž1986., 81᎐84. 3. H. Bor, On the relative strength of two absolute summability methods, Proc. Amer. Math. Soc. 113 Ž1991., 1009᎐1015. 4. H. Bor and B. Thorpe, A note on two absolute summability methods, Analysis 12 Ž1992., 1᎐3. 5. T. M. Flett, On an extension of absolute summability and theorems of Littlewood and Paley, Proc. London Math. Soc. 7 Ž1957., 113᎐141. 6. M. A. Sarigol, ¨ On absolute summability factors, Comment. Math. Prace Mat. 31 Ž1991., 157᎐163. 7. M. A. Sarigol, ¨ On absolute normal matrix summability methods, Glas. Mat. 28 Ž1993., 53᎐60.