Mathematica Slovaca

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Mathematica Slovaca

Billy E. Rhoades

Absolute comparison theorems for double weighted mean and double Cesàro means

Mathematica Slovaca, Vol. 48 (1998), No. 3, 285--301 Persistent URL: http://dml.cz/dmlcz/136728

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Mathemotica Slovaca

ABSOLUTE COMPARISON THEOREMS FOR DOUBLE WEIGHTED MEAN

AND DOUBLE CESARO MEANS

B. E. RHOADES (Communicated by Eubica Hold)

ABSTRACT. In a recent paper [SARIGOL, M. A.—BOR, H.: On two summability methods, Math. Slovaca 43 (1993), 317-325] the authors showed that the Cesaro means of order a are absolutely k-stronger than weighted means satisfying the condition P

n

= O(n

a

p

n

), 0 < a < 1. It is the purpose of this paper to extend this result to double summability.

Let {s

jk

} denote a double sequence. The ran-term of the (IV, p^.)-transform of the sequence {s

jk

} is defined by

1 m n

T • = — r r ^{D s}

mn ' p / ^ / ^rii^jj 5 mn i=QJ=Q

where

7Ti n

imn '~ / ^ / v Pij ' i=0 j=0

The ran-term of the (C, a, /?)-transform of a sequence {Sj

k

} is defined by

i 771 n

mn ' TTct Tpl3 £-< £-s ^m-i^n-j *ij »

^m11"*1 i=0 j=0

where

E: "' V « )

A double sequence {p^} is factorable if there exist single sequences {p

{

} and {q-} such that p

{

- = p

{

q^. We restrict our attention to weighted mean methods

AMS S u b j e c t C l a s s i f i c a t i o n (1991): Primary 40D25, 40G04, 40G99.

Key w o r d s : double Cesaro matrix, double weighted mean matrix, absolute inclusion.

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generated by factorable sequences, since it was shown in [1] that the condition of being factorable is necessary in order to find the inverse of the transform. For any double sequence {u

^{j

}, A

¹⁰

u

^{j

:= u

^{j

- u

^i+lj

, A

^{0 1}

t i

^y

:= u

^{j

- u

^ij+1

, and

A

n

^u

ij '•=

^u

ij ~

^u

U+i "

^u

i+u +

^u

i+U+i'

THEOREM 1. Let 0 < a, (3 < 1. If {p^} is factorable, nondecreasing, and

P

mn - 0 ( 1 ) , (1)

p

^mn

(m + l)"(n + l)f>

then \N,Pij\k summability implies |C,a,(3\

^k

summability, k>l.

P r o o f . Since {Pij} is factorable, we shall assume that we may write p^

in the form p

^{

q-, where {p

^{

} and {q-} are positive nondecreasing sequences satisfying the conditions of the theorem. Then

T

тnn

1 m n

m^n i=o j=0

Let {s

^{j

} be absolutely fc-summable by the weighted mean method defined by (2). This means that

oo oo

V2^(

^{m n}

)^|

^{A l l}

r

^{m n}

|

^{f e}

<oo.

m=0 n=0

We shall now use (1) to obtain explicit expressions for the a-• in terms of the T-.. Using (2) with m = 0 we obtain

T0n ~ Q 2-s^q3^S0j'

V» 3=0

Qn^TOn ~ Qn-l^TO^yn-l⁼ Qn^SOn ' (³)

Using (2) with m = 1, n > 1, yields 1 *

ⁿ

In — p Q / J / ^Pi"iSii ' l^n i=o j=0

1

~l«?n~l„ - Q„-l~l,„-l) = E I W i n •

i=0

Pl\QnTln ~ Qn-lTl,n-l) . / • \ • "T^

"^! = P0⁵0n +PlSln = (PO +Pl)^SOn +PlZ^^aOk '

q™ k=0

286

(4)

ABSOLUTE COMPARISON THEOREMS FOR DOUBLE WEIGHTED MEAN Using (3),

Pi_Z

^a ^lfc

_Pi(QnTщ-Qn-iTi,n-i) PЛQ

n

Ton-Qn-iTo,n-i)

k=0

= P i

= Pl

Qn 1n ____(rp _^— \ i ~ _ ____(rr _ T ". _ T

n V^ln ^l, n -l/ "*" ^l, n -l⁰ ^ O n ^O.n-lJ ^ 0 , n -l

"n ^-n

^ n

L«n

^{^}

^{д T - Д T}

^{l l}^{^ n -}l ^ Ю ^ n -l

Thus

_ P i

_ Pi

-^2-A T — A T

п

-ni-^o.n-i ^ю-^o.n-i Qл

И n Qn

Qn-i

Яn-l

-Чl-l0,n-2 + ^10I0,n-2

Qn-i

q

л

A T -п-

í

o , n - i

q

.

L n ^n—1

Similarly, for ra > 1, Qi

m l

q_

Using (2) for ra,n > 1,

-A,,T

^ - l l - M ) , n - 2 + ^l l-MЗ,n-2

Pm-1

l l^ m -1, 0 Pm-1 " Д ц - S n - 2 , 0 + Пľ m - 2 , 0

n

PmQn^Tmn ~~^Pm-lQn^Pm-l,n⁼ _\__PmQj^Smj ' j=0

n -1

PmQn-l^Pm,n-l ~~^Pm-lQ n-1* m-1 ,n-l ~~ ____Pm⁽lj^Smj ' j=0

and hence

(4)

(5)

POT — P O T —PO T

m^nmn m — l^ n m—l,n m ^ n — 1 m,n —1

+ ^ m - l ^ n - l ^ m - l , n - l⁼ PmQnSmn '

_ _ _ _ _ n ~ (^Pm~Pm\Qn^T (Qn~~Qn\^Pm^T PmQn ^ V Pm ) <ln " " ^ V ?» J Pm ^ ^

_ _ m^L_ _ A T _ _ _ . A T^m A T 4 - T

— ~~~~7~^-ll^m-l,n-l „ ^ Ol^ m -l. n -1^w ^ l O ^ m - l . n - l ~ ^ m -l, n -l * VmVn ^n -^m

(5)

A ( m-lVlA^T 1 . ___A r

"mn ^ 1 1 l^{v Q} ^ l lⁱm -2, n -2 I + ^^Al l^Jm - 2 ,ⁿ- l

\ -^m—l^n—1 / ^n Qn-l

a \l~^m-2yn-2 + J" ^ 1 1 ^ - 1 ^ -2 (⁶)

* n - l /'m

•*m-l

P m - 1 ^ Ц - * m -2 ,n -2 + ^ l l - S n -2 ,n -2

In a similar manner it can be shown that a - - J - 2 - A T

11

^

^lir

°°- ⁽⁷⁾

Let JJJ^ denote the mn-term of the (C, a, /?)-transform in terms of {mna

^mn

} where a

^{m n}

is expressed in terms of T

^mn

. Then, to prove the comparison, it will be sufficient to show that

oo oo

m=l n=l

Using (4) -(7),

-. m n

t<mn ~ roc r?ß jLRa pP ^fZ^Êm-lÊn-j^tJâij

ĽJmĽJn i=0 j=0 - m n

JZ/m£jn i=l j=l

E^EÍ

m . П

-%-#

j=l i-2j=l

E

m-\

^E

t\

^a

ii + f^E

^m

Z

^l

Xz

^1j

ja

^lj

Л-ү^E^zЏt-ÌЩi

3=2 i=2

+ E E ^ Г -

m n

¹ Æ ^

i=2 j=2

= w

^г

+ w

²

-Ь г 0 з + г0

⁴

, say.

KL

Oü Oü - OO ŁЛJ

E E i w ^fc = od) E E(™г ^fc_1 = °м

m=2 n=2 m=2 n=2

' _ . -,

« J - I -«І

1 V E

^a

~

^l

E

⁰

'

¹

i (^

m n j=2^x

ІЛ T -

³

~ A T

?,•

Л l l i

° . ; - i ^ -Чi-o„-

2

+"-Ц-"o_-2

(6)

ABSOLUTE COMPARISON THEOREMS FO R DO UBLE WEIGHTED MEAN :_= 1v

2

l + ™

2

2 .

S a

y •

w

2i = pE

a

EP

ґ l m n

P

¹

E£\

a

E

^t

m n

A______L

r

* E

^am

EÍ

V к ^ Æ д т -TГE

⁰

-

¹

!^- Z^^n-jJ o. --11-Чi-i 2-j^n-jЗ

^Q

J = 2^Уí j = 2 * J - -

~ - l l -f0 , j - 2

^Д„г

^{0 t}

._

¹

-__. í:_^д

^{1 1}

Гoo

n - 1

,__.

+ £ 0 - Є J - (i + D^K-i) f A i Ä . - i

i=2

V

í

= u _ n + ^212 +

W

213 • say.

OO OO

££^Kui

m = 2 n = 2

0 0 0 0

= £ £ —

---' ---' mn

m = 2 n = 2

= 0(1) £ m-*- ¹ f V ¹ (-^-)ViiTo,n-iЃ

m=2 n=2 \

П

ЧnS

0 0

= 0 ( l ) _ Г n * -

¹

| Д

^{1 1}

Г o

^{i f}

_

¹

| * - - 0 ( l ) .

Л___________

^{д T}

I

PiE^EІ q

^{n n}

^-Ҷ

n = 2

00 00 00 00

_____ -_--- ran'

^{2 1 2}

' ^ ' ---' ---' ran

m-=2 n = 2 m = 2 n = 2

0 0 0 0

jTta—1 TTI/3 — 1 |k

Ьт-1^Ьп-2 A T

Ecx^E0 ^-11-00 m n

= 0 ( l ) £ £ ( m n ) - * -

¹

- - 0 ( l ) .

m = 2 n = 2

From [2; p. 320], jE

⁰ⁿ

Z) " --t-.-iO" + 1) = i - ? „ l j - - - f t - i • Write __

¹³

=

^2131 + ^2132*

Using Holder's inequality and the results on the last line of page 322 and on page 323 of [2],

CO CO

£ £ ^ n n K i 3 i l ran'

m = 2 n = 2 co co

ran I E

^a

E@

m = 2 n = 2 ' m n _/__2 pa-l n - 1

-- -- . E

^a

~

^L

?":. O-

= ^

1

) X , Z ^ í~7{ vaғ ß _ C ^ n - j - Г

Л

И

Г

0 J - l

Í J

(7)

n - 1

=O(Df „.-+-_>-> i Y . i C l i ( ^ ) .A..JV.I

m=2 n=2 L ™ j=2 \

y

_ /

n - 1 ^{I fc-1}

ÊßJ2\ E nч\

,ⁿ j=2

0 0

/ iO \

k

1 °°

-owEЃ

²

?) i^

^w

-.i'--f---r E

7 = 2 ^ ^ / ^ 7 + 11 П=7-ł- І = 2

OO

j=2^V ^ / 00

= o(i)X)j*

+/í

-

2

|A

11

r

()ií

._

1

**|*--o(i).**

J = 2

k 1 ^ l-^n-jl

(-&.)*- ¹ -4+i ^ni? «

- l ) - 2

OO OO

Z-_ )__. ^ K l 3 2 І mn

^ҳ

m=2n=2

OO 0 0

m = 2 n = 2

00 00 1 z ? Q - l л - l r)

^

⁰

^

¹

) _C Zľ ^ j ľ^/ З Zľ

^{£ ;}

n - j - l - /

A

l l

T

0 , j - l

n j _ - 2

= 0(1) ^ m - * - ^ " "

1

m = 2 n = 2

áE-tí-ífO'.-..^-,.'

^ n j=2 X •? /

n - 1 1

I Г E ^ -

¹

L n j = 2 J

i / e - l

0 0 _ ţ _ "^{9 _ 1}

=

⁰

(D_;(|) I A , , ^ . / _; ^ 1

j = 2^V ^ ' n=j+l^n£jn

OO

= O(l)Y,J

^Pk

**\*nT**

⁰

,

^j

-

¹

\

^k

r

¹

=0(1).

J = 2

OO OO

Z_/ Z_v ШП

m = 2 n = 2

21321

rnn I E

a

E

ß

m=2n=2 ' rn n j=2

E J Q - 1 n

^(DEľ àtóD^v

(8)

ABSOLUTE COMPARISON THEOREMS FOR DOUBLE WEIGHTED MEAN

00 00 r n

^ D E ^ E - - ¹ ÀE^-]i

^д

"

^г

oj-2І ^å

L n j=2

m = 2 n = 2

00 Eß-\

Eß _ / _ / ^n-j - n j=2

k-1

O(l)J2j ^k \A ⁿ T ^0J _ ² \^-^

j=2

00

n=j

= O(l)"£j

k

-

1

\A

11

T

0J

_

2

\

k

= O(l).

3=1

Since w

3

is iu

2

with the roles of p

i

and g. interchanged, it follows that

00 00

EEiKi = o(i).*

ra=2n=2

From (6), we can write w

4

= w

41

+i_>

42

+i_>

43

+ w

44

+ iU

45

+ w

46

.

»« = ^ E E ^ r 4 ^ ] A ⁿ ( 5 ^ A ^{1 1} T ⁱ _ ^{2 J} _ ² )

^m^ni=2j=2 \ ^i-l^-j-1 /

1

m [ n P O

- • ^ - f T - ^ " ' E ^ n l ^ A n T ,,•* ²

77-Q 771/3 ____-/ " - - * _____/J ~ - j ~ a < 11 i-2j-2

^ m ^

i = 2

[ j = 2 n - l ^ J - l

n

P O

n

PO

j=2 ^1-1^3 j=2 - ^ t y - l

n

P O

„ _ o - ^ * 7

n - S - i . j - 2

1 m

E

a

E

ß Z^^rn-г

^rn^n i = 2

n

F

ß-l

P

i-lQl д

^T

^ n - 2 — Г " ^ l l ^ i - 2 , 0 Pi-lЧl

"Pj-lQn

Pi-lЯn A^T _n-Ч-г.n- _i

n - l

+ £«-• + D-^zj-i - j^:])5 ^r T ^ÍA n^-2.-i

. = 2 ^ - - ^

_ 9/V!

9-1

____-! A T | — г ^ n л P û

^T

Z j C /

n-2 „

ⁿ

^1_-

£

І-1,0 +

Ü 1 1 І

-

E

a

ғß

p_

qi

„ .-x,»

^p

.

^g

_ -"ll-S-l.n-l

E(( j +U-&.-1 - jIe^^r^,,/

•í=2 '^{J J}

ž ^ - o Í F ^ r ^ ^ ^ - A , ^

²⁰

^-YÍE°-

^IP

^&

n — 2 1 / __ m—i tr\^Q 11 i—_,U X _ .

V , = 9 ťi-1*! ;_o

A T

(9)

E°LE*

i=2 \i=2 ^i-l^i

m p Q \

Z^,írjm-i jy a L* n1i - l j - l

t=2 ^ ' >

(

^m

^{P O}

^m

j=>n '

E^-iT^Ani;-..»-! - E^-i^n-^-i.n-i

i=2 ^ . - l * n i = 2 />.«„

2E

^e>

~\\ 2E

^a

~

^{1 F>1}

®

¹

A T m

^P

™Q\ A T

n

"

²

V

^TO

"

²

Pi?i

^{n 0 0}

p

^ro

?i

^{A l l T}

™ - i . °

+ E («+-)-^-î-i - ^ r - ¹ . ) ^ ^ ^ ^o )

n - 1

+ E(0' + ¹ ) ^ : j - i - j ^ : j ) x

j=2

(

?a

-г

^P

гQ.

pm

Qj

V " i - - p űf. n U,j — 1 p ^ 11 m—l,j — 1

P I Í J

Pm^i

7П —1

P O \

E «* + D-^-U - i-SS) -^A

^{1 1}

T

ⁱ

_

^{1 J}

_

¹

)

ž=2 * ^ Í /

- n(2E

^a

~

¹

-

ⁿ

A T m

^m

"

ⁿ

A T

" l ^ -

^{2 Pl}

q

^{n A l l I}

° ' " -

^{1 m}

P

^m

«7

^{n A}

i i

^r

» - i . » - i

^ - ? PiO \

+ E («+-)-%-.-i - ^r- ¹ .) v r A . . ^ . . ^ . . )

i=2 ^ ^ n /

™411 + ™412 + ^413 + ™414 + ™415 + ™416 + ™417 + ™418 + ™419 »

say.

CXJ o o o o o o

**E E ^ K П І * = O ( D _ : E ^**

m=2 n=2 m = 2 n = 2

oo oo

jp/3—1 rpa—1

^n-2-^m-2

Et,E

^ß

m n

0(1) ^^(mn)-"-

¹

=0(1).

E E ^{m n} Kis

m=2 n=2 oo oo

-ОД ľľi

v 7

-—•< -—' mn

^{m=2 n=2}

m=2n=2

^flo

n-2 ¹

P

m-^n ^тn

n~z m-ЛŁЛ T

(10)

ABSOLUTE COMPARISON THEOREMS FOR DOUBLE WEIGHTED MEAN

00

°° / P \

k

**= 0 ( 1 ) - > - * " ' E m W ^ ) |A u T m _ li0** |*

_, o »-.-. o \ "m /

n=z m=2

00

= O ( l ) ^ m -

1

| A

1 1

T

m

_

l i 0

| ' = = O ( l ) .

m=2

>

^m

-i + ^ m - f - i , we may on

m=2

Using the identity (t + l)E^\_

x

- iE

m

~_\ = - _ E _ j write w

il3

= w

4131

+w

il32

.

Using Holder's inequality and the results on the last line of page 322 and page 323 of [2],

OO CO

£ £m^Ki3il

m=2 n=2

00 00 rpP-1 m—1 p i k

=°w £ £ ^ _ ^ E H ^ - ^ A ^ J

m=2n=2 ^m^n i=2 y% '

00 00 r m — 1 / ' P \ k

= 0(1) J > - * - " > - ' ^ - E l ^ - ^a « l ( ? ) l ^A n ^T i-i/

n=2 m=2 L m i=2 \ ^ / r m —1

x _^£i^r4

L "» i

= 2

^WzZyj

1

) l

A

n

r

i-i,ol

(Ea

xfc_i £ _-___*

i = 2

V

**^ * ' < A + 1 ! m = i + l**

m

= o(i)_rz

^{f c +}

«-

²

|A

ⁿ

r

^<

_

^{l i 0}

|*--o(i).

*—o

X

- A:—-1

i=2

00 0 0 00 0 0

£ £m^Ki32l

m=2 n=2

00 0 0

=0(1) £ £ -

m=2n=2

тpß-1 m - 1 p

^ n - 2 V~> тpa-1 £ _ д rp

E

cc

E

ß Z^^m-i-lp.^ll^i-l;

^m^n

І—2 г

**^ ' ._-_, Z-_/ mn E^EP ---* rn-г-i**

^p

11 t-i,u m=2n=2 ^ ^ n І__2 ^

г

'

00 00 r m —1 / P \ ^

o(D £n-*-> _г ҷ~

¹

_ғ £ ^ - І - I ( ? ) IДЦÏІ-1,0

n=2 m=2

^{l m}

i=2

Ч

^

г /

[ ^ õ " ___, ^m-i-l ^m—1

m

i=2 00 / p \k

= 0 ( 1 ) g U ) ^|A " ^T ^.o

X

- i f c - 1

00 E ^ a - l

|k V ^ ^m-i-l

1

---< m _ 7 "

m=i+l

m

(11)

CO

= o(i)E»"

^a

*"

¹

l

^A

n

^r

«-i/ = o(i).

i=2

Writing w

414

= w

4141

+ w

4142

, we have

CO o o

E E —

_—/ Z—/ rnn m=2n=2

00 00

m n

| U

'

4 1

«

l

тpa—1 n — 1 >-»

-c-V>-._o x:—> . д _ o . ЦГ7-

: 0 ( 1 ) Г Г І _ _ . У ( . # ? ] _ Д

T

^

X

! __-< _L_

m n

ßaßß 2_Л •/-t V-.J -. --ll-Чj-l

nг n j_-2 .7 ra=2n=2

00 00

00 00 r n - 1 ,-f) v fc

m=2 n=2 L n j___2 \ ^J / n - 1

i>Ei^-1i

n j=2

lk- 1

= ° W _ ( f ) ^.^-.řjp-rjzT _ _pj

Í=2^V ^ ' v^j+J n=j+l^UtL,n 00

= 0 ( l ) _ C - *

⁺

'

^,

"

²

l

^A

i i - - j - i l * = 0(-)-

J'=2

2_/ _ L ^ Í _ K І 4 2 І ran'

m=2 n=2 00 00

m=2n=2

00 00

- °(-) L Z - m n p;a

E

i3 ___•

E

n-j-i

q3 A

n

T

o,j-i

m n j=2 3 n—1 / r\ \ к oo OO r П - l У І - J ч k

=o(D _ »->- - „- U - ^EEЦ%) IД, A . . , 1

m=2 n=2 L n J=_2 \ *J /

г n - 1

x -І-VPЛ-

¹

X

[__;£ z L / ^ n - j - l

= 0(l)_(5í) |_„T ^tJ . ¹ f _ % ^

j=2 \ ^ / n = j + l^{П j C /}"

n j=²

k-1

= O ( l ) _ _ j

w

-

1

| A

1 1

T

0 i i

_

1

| * - - O ( l ) .

J=2

(12)

ABSOLUTE COMPARISON THEOREMS FOR DOUBLE WEIGHTED MEAN

Writing w

415

= tw

4151

+ w

il52

,

W 41511

00 00

EE—,

._—v z__/ run'

m = 2 n = 2 00 00

= 0(1) 2__. \^—\

E

a

E

ßLj(~i

E

ri-j)

D a

,

A

ll

T

m-l,j-l

n - 1

771 = 2 n = 2

00 00

n j= 2

n - 1

• • - ^ - 2 ^ m Q ji

joV

fc

OO OO r П — 1 / IГЛ ч /Ç

=0(1) £ ™ » - £ „ - U £ \ E І - Ą\{m IДÆ.,,.,1

m = 2 n = 2 L n j=2 \ *J /

n - 1

-^E^n-jl

n j = 2

J c - 1

00 | i т i / 3 - 2 | OO OO / Ą(} \ & OO Iттip — -

= 0 ( i , £ ^m ' - £ ( ^ i ) |д

^{l Љ}

_,

^j

_

¹

|' -

^ғ

l— £ g

m = 2 j=2Ч ^ ' У j+l) n=j + l ПĽJn 00 00

= 0(1) **£ m*"**

1

E^

+/3

"

2

l

A

п

Гm

-i.i-il

fe

= ^

1

) •

m = 2 j=2

1 I 1 — ^ / J - i " ^ m Q j

EE —

Z—/ Z—/ 77277, m = 2 n = 2

00 00

= °(

¹

) z2 z J 77m\E

^a

EP ^

^En

~3-

¹

p^/

^A

n

^T

m-ij-i

m = 2 n = 2 ' m^n j = 2 - ^ m ^ j 00 00 r n — 1 / r\ \ &

m = 2 n = 2 L n j = 2 \ J J

'QЛ*

n - l 1 \ ~ Ei/3-1

77./З Z^^n-j-1 n j=2

00 ^ g / з - i

k-1

00 /1) \^к r^н

= 0 ( l ) £ m - ' £ ( Ş ) iд

¹¹

г..

^li)

.

^I

ř £ - ^ j i

m = 2 j = 2 V^Ч3 ' n=j+l ПĽJn

= 0(l)X/

^{/ 3}

-

¹

|AnT

^m

_

^{l i j}

_

¹

|

^{f c}

= 0(l).

3=2

Writing w

^{4 1 6}

= u>

^{4 1 6 1}

+ ty

^{4 1 6 2}

+ w

^{4 1 6 3}

+ w

^il64

,

(13)

E E

^{m n}

1^41611

m=2 n=2

oo oo i m—1 n—1 p r^

=

W J-

^x

V \~ijE

^a

-

²

E

^p

-

²

-Í-í--A T Z—• Z-^ mn\ E

^a

Efi -^—' --

^m

~

^{l n}

~^va-

^{u l}

~

^l

^~

m=2n=2 \J~JmJ-Jn i=2 j=2 ^3^3

00

°° i r i

^{m _ l n}

"

¹

fijPQ \

^k

1

m = 2 n = 2 L ^ m ^ n i = 2 j=2 V * ^ 7 J

m—1n—1

1 V ^ V ^ \Ea~2E(3~2\ Ea E& •* J ' m —* n""i '

^чÊŽ( ^í ^ Ł )'i-. Д-. ^J -.i ^, x

ѓ = 2 j = 2Ч ľгЧ3 '

m~n І—2 j=2

oo oo

k-l

x 1 y> y^ 1-^-i-^n-?!

(-sa-i-í+i)- ^{1 m} =r ⁺ m=7ti ™~\K*

^oíDEE^^^iДц^-i.-ii fc ( )"

f a ^ f c - l i=2 j = 2

oo oo

(ІV)

=od) E E ^)*"

1

д

п

г

-i j-i i = °(-) •

i=2 j = 2

mn'

7 n=2 n = 2 oo oo

EE

77г=2n=2 7ПП

J_

7ПП

m—1n—1

m

E°

E E -

i m—i n—i /PO \

=

Yl YmljTaRfi Z2 2Z^~

lE

m-i)

E

n-j-l[ Y^T )

A

ll

T

i-l,j-l

i n i=2 j = 2 \r3^3 /

m-ln-1 (iPQ.\k 1

~^š E ^ I C - i l C j - i h r ¹ l ^A ii ^T i-i,j-il ^fc

i n i=2 j=2 V y%HJ ' J

m—1n—1

7~~3 Z-*, Z s l ^ m - i l ^ n - j - l i n i=2 j=2

o w E E ( ^ ) I^..Í'.-. ^J -.I'(^V -- --

i = 2 j = 2V %1 ' v-^i+11 m = i + l n = j + l

fc-1

OO OO I ITiQ — 2 i jT.,-3 — 1

.-^-ÍI-CU

OO OO

mnE^Eß

= °( ¹ )EE ⁱ ' ^C+a " ² ^- ¹ |A ¹¹ T ⁱ _ ^1J _ ¹ | ^fc = 0(l).

i=2 j = 2

(14)

oo oo

Z> 2_, ^ K l 6 s l

m = 2 n = 2 oo oo

= E E —

-^ -^ mn

m=2n=2 ^{0 0 0 0}

E E '

m = 2 n = 2

mn

m - l n - l ^ fP.Q.\

Z?QJ?/3 ž ^ Z^

^E

n\-i-l("J

^E

n-j)[ "TT" )\l

^T

i-lJ~l

^ m ^ n i=2 j=2 ^ F%H3 '

=tf J2 .S^m-i-ll^n-jlí-^^) l A ll T i-l,j-l

Jn i=2 j=2 V y%HJ '

r m—1 n—1 m — 1 n — 1 k-1 1 V ^ V ^ pOL-l i rp/3-2 J?a p/3 Z - / .Z_w m - i - 1 l ^ n - j L m ^ n i = 2 i = 2

00 00 x • p /") ч /e 00 00 трос—l i r-i/3—2

= 0

w _ _ L h z i--ii

^r

. -u-ii ,

^p

/. 4

^fc

_i E E

^mm

nE

^a

E^

i = 2 j = 24 ť t 4- ? / УЬ}+1) m = i + l n = j + l т П£ /т ^ п

OO OO

= а д E E j*-"-

²

.*--

¹

1 ДцГ,.! ^ i

^fc

= 0(1).

V — 9 1 — 9

i=2 j=2

0 0 0 0

E EmT^™

⁴

^!'

m = 2 n = 2

OO OO

= E E

m = 2 n = 2

mn

0 0 0 0

= E E

m = 2 n = 2

mn

m—l n — 1 / pt) \

_J r r p a - i Ei/3-1

(Í±1±\A T

^ JPP / L / L V i - i V i - i „ 0 j ^ n - s - i j - i

^ m ^ n i^{= 2} j=2 ^ *^iH3 ' m — l n — 1 / P f) \ ^

/?«/?/? Z ^ 2-_,

^E

m-i-l

^E

n-j-l [ "^7~ ) I

^A

ll

^T

i-l, j -l

-^m^n i=2 j=2 ^ F t^ ' r m —1 n — 1

1 - . — % - _ — - . o -. m — 1 n — 1 1 V ^ V ^ J ? « - l /ľ-Æ-i fj<* fîß ' J ' ..* m - i - 1 n - 7 - l - m n ѓ=2 j=2

X

OQ 00 , -p r\ \k 00 00 i^c-—1 / ?l^^—l

^ Í ) E E | ..vw-f E E ^ f g f ¹

i = 2 j=2 V ^ ' m = i + l n = j + l mUrjmIljn 0 0 0 0

= O f l ) E E i * " -

1

j * " -

1

1 A ^ , ^ |

f c

= 0(1).

i=2 j=2

0 0 0 0

EE^Kiri*

m = 2 n = 2 т = 2 п = 2

ОО ОО у ^ ч

=°(!) Е Е ш Е^

^ПЕ

ЩТГ»

^Т

<>^

т=2п=2 ^т^п \ *п /

297

(15)

00 0 0

= 0(l)X;я.-

^fc

-

¹

E

ⁿ

*"

¹

l

^A

иГo.„-il

^fc

=

m = 2 n = 2

0(1).

00 00

EE^Kisi*

m = 2 n = 2

III £é II _i

У^У^J 1

^mnP

mQn

^{A T}

Z - Ѓ Z - / JПП Fa Fß V Q n m- 1 >n- 1 m = 2 n = 2 ^m^n ym^n

00 00

<_XJ OU

=

° ( ! ) E E ( ^ )

^{f e}

"

¹

|

^A

l i

^{r m}

- l , n - l |

^{f c}

= 0 ( D •

m = 2 n = 2

Writing w

_'419⁴¹⁹_—

= w

^ 4 1 9 1 + ^ 4 1 9 2 ⁴¹⁹¹

+ w

^41s

0 0 0 0

E E i^Kml*

m = 2 n = 2 0 0 0 0

EE * _{' mn}

m=2 n = 2 0 0 0 0

1 m _ 1 PП

W iE

^a

~

²

)

^{{ n}

A T

zpa. ipß Z — / ^ m—i) ) n 11 i — l , n — 1

^m^n i=2 F i Ч n

00 00 г m—1 • . p ч k

° « E E ^ ^ - E i ^ i í ^ ) i ^A п-u„-.i ^fc

TT7-2 rx=2 L m i=2 Ч г ' J

* v . _ 1

m = 2 n = 2

X -jfc-1 m — 1

~~~~ ___/ I m-i\

L m ť = 2

00 °° / ň P \^k °° IF^ - "²

n = 2 ť = 2^{V F%} ' ^ t + 11 m = i + l^m 00 00

o(i)E»"

¹

E

^ť

*

^+a

"

^a

i

^A

ii

³

i-i.«-ii* =

n = 2 i = 2

= 0(1).

00 0 0

m = 2 n = 2 00 00

= E E —

-—< -—' mn

m = 2 n = 2 00 00

1^{m _ 1} PO 1 \^A 1-a—1 ^ⁿ A T zpec fPfi / J m — i—l pQ ^ ~ l l ^ - i — l , n —1

nx n- i__;2

00 00 r m—l / p \ k

°«EEi F E C L M I^I-U^I*

„, o ^ o L m • o \ - * / m,=zz n = z t=_J

7n —1

£~T ^ _ , | ^ m - i - l

- m 7=2 m = 2 n = 2

X

-1 k-1

(16)

OO T 7 I Q - 1

00 00 / p \ k 00

^ ( D E ^ E Í ^ J ^11^-1.-1 \ k E

n = 2 i=2 ^ l ' m = t + l

771'

^ m - i - 1

mEZ

00 00

= 0(1) _ Г n "

1

_ Г ť*-"

¹

1 Д

1 1

Г

ł

_

^{l iП}

_

1

|

fc

= 0(1).

n = 2 i=2

mn

l

EE-

Z—/ Z-J m\

m=2n=2

00 00 1 m n r\

Z-; mn\E

a

E

ß

---' ---'

m _ i n

~

j

Q-

n i

"

2

'-

7

'"

1

m = 2 n = 2 ' rn n i=2 j=2 3

o o o o г m n / Л \ fc -1

E E " ~ ; \

E

a

E

ß E E

E

m-i

E

n-j [ -— ) IДцT І-г.j-ll

m = 2 n = 2 L m n i = 2 j = 2 W J / J

-owEEÍ?) i-un-,...!' __ E

t = 2 j = 2^V -*⁷ m = t + l n = j 4 l 00 0 0

=o(i) E E

^{r 1}

^

^fc/3

"

¹

1

^A

n

^T

i-2,i-i i

^fc

= o(i) •

ť = 2 j = 2

m n - fc—1

1 " p " p Fľ

⁰

*

^-1

ғ

^ß

~

l pa fiß Z - v ZІ_-< m-i^n-j

^m^n i=2 j=2

00 00 zpa—1 тpß—1 zpa—L T?£

•^m—in—j

mnE

a

E?

ou ou

EE-

Z__/ Z—v 771.

m = 2 n = 2 0 0 0 0

mn* \w.

⁴³¹

m = 2 n = 2 0 0 0 0

E Г ^ 1 1 " ^ ^ p g - l ^ - ^ H д j - Zs mn \E

~m^n i=2 j=2 ^a

E

ß

---' -^ ~

{

~

j

O2 л -- "*

²

'

^j

~

pa-ljpß-lQjЫ

«i-l

0 0 0 0 - m n / O • \

= E E ^ - _ - - j E E

£

m - i

£

« - i ( g r r ) i

A

"

T

i-2,j-2i*

»-~—o*-. — o L-^TTI. n i — o „•— o \ J 1 /

m = 2 n = 2 L m n г =2 j = 2

---ľľrt"

i ? a Z7/3 Z ^ ZL_/ ^ m - г ^ n - j m n ¹

~- n ѓ = 2 j = 2

X k-1

z 00 00 n a - 1 pß — 1

iд т I* V V

ш

~

{ n

~

j

l^и

2

ѓ-2,j-2І 2-é Z^ mnE

a

E

ß

m=i+ln=j+l

^mПГy

m^n

00 00 / r\

0 « E E | ^

. = 2 j=2 V 1 X

0 0 0 0

:

o(i) E E

^ť

" V "

¹

i

^A

n

^T

i-2,,-2i

^fc

= 0(1) •

t = 2 j = 2

(17)

00 00

££db к ⁴ i

m = 2 n = 2 00 00

= £ £ —

--- -—' mn

m = 2 n = 2 00 00

1 " P "Г^ pa-1 pß-1^гi д -1

EaEß Z^Zs^m-i^n-j p/^ll-Li-lj-ï

m n І=2 j=2^г

0 0

°° 1 Г 1

m n

ҐP\

k

=

Z^Z^Ш E

^a

E

^ß

^ ^

^E

m-i

^E

n-j ( Г

1

) I

A

11

Г

І-1J-2І

X

m = 2 n =2 L-^m^n i = 2 j = 2 Ч г 7 J

m n

-ja Eß -—' - — ' m - г ^ n - j Ľ ^ m ^ n i—ⁿ • «

= t f ( D £ £ ( £ ) lДцЗl-u-21* £ E

i = 2 j=2 V г 7 m = i +l n =j+

L m n ѓ⁼2 j = 2 00 00 iľia—1 гpß—1

.

lл

mnE

a

E%

= 7+1. m n k-1

00 00

=o(i) £ £ i ^kß -Ч-

¹

1ЛÆ,,,._/ = o(i)

i = 2 j = 2

£ £ ^ K ⁵ I *

m = 2 n = 2

00 00 m n —1

= Y^ Y^ — 1 Y

>

Y^E

,a

~

1

/~

/3

~

1

^ A T

Zs Z-J mn EaEP2-j2-*t m-i n-j v l i i - 2 , j - m = 2 n = 2 m n i = 2 j=2 l~L

co oo r m n / P \ ^

~ .2-/ Z _ / ~ ~ ~ -?a ir/3 .2-/.__/ Em-i&n-j l p ) lZ" l lii - 2 , j m = 2 n = 2 ^ m ^ n i=2 j = 2 \ - ~ t - i /

= 0(1)

0O OO / n \ f e 00 oo

££(£7) I-..W.I* £ £

i = 2 j=2 X %~1 ' m = i + l n = j +

00 00

=o(i)EE

^ť

*

^/,

"

¹

r

¹

iA

¹¹

т

ⁱ

.

^{2 i J}

..i

i=2 j=2

m n

IPa z?/3 Z_/ Z-/ тn-г n-э m n i=2 j=2

00 00 ï?a—1 тpß—1 Ь^m-i n-j

! ł + lя_^{J + 1} ^ m ^

0(1).

X

Л -l

EE>«i*

m = 2 n = 2

oo oo m n

=

L L т а RC из E E ^ m - i ^ n - i A ц T i - г j -

o o -^Ł-^/r r i -^{L V}n .• o >; o

m = 2 n = 2 m n i__2 j=2

(18)

00 00 r m n

ž ^ Z.v ^Ц E

a

E

^ß

-' - ^ ^™'

ⁿ

~

^j^uTi

~

2

>з-

m=2 n=2 m n 2=2 j = 2

X 1

Г д

⁰

-

¹

^-

¹

E^a E@ Ć-J Z-4 ™~гⁿ~3 m n i = 2 j=2

тfc-1

00 00 00 00 poc—1 pß — 1

=o(i)£ү> ¹¹ т ⁱ _ ¹⁾ ._ ¹ **i* _г x; — ^-**

г=2 j=2 00 00

m=i+l n=j+l

mnE

^a

Ef?

=o(i)^2Y,(

ⁱ

Jr

¹

\\

¹

T

ⁱ

_

^lij

_

¹

\

^k

= o(i).

t=2 j=2

There is no need to consider the comparison for either a or /? > 1, since those cases come as consequences of the translativity of inclusion and the fact that the well-known result of Flett comparing absolutely the Cesaro matrices of

orders 7 and 5 readily extends to double summability. •

R E F E R E N C E S

[1] MORICZ, F—R H O A D E S , B. E.: DouЫe weighted mean methods equivalent to (C, 1,1), Publ. Math. Debrecen 4 7 (1995), 29-64.

[2] S A R I G Ö L , M. A . — B O R , H.: On two summability methods, M a t h . Slovaca 4 3 (1993), 317-325.

Received August 31, 1994 Department of Mathematics

Indiana University

Bloomington, Indiana 4^405-5701 U. S. A.

E-mail: rhoades@indiana.edu

Mathematica Slovaca