Logarithmic matrix transformations into Gw

PII. S0161171204303418 http://ijmms.hindawi.com © Hindawi Publishing Corp.

LOGARITHMIC MATRIX TRANSFORMATIONS INTO

G

MULATU LEMMA

Received 10 March 2003 and in revised form 1 July 2003

We introduced the logarithmic matrixLtand studied it as mappings intoandGin 1998 and 2000, respectively. In this paper, we studyLtas mappings intoGw.

2000 Mathematics Subject Classification: 40A05, 40C05.

1. Introduction. The logarithmic power series method of summability [1], denoted byL, is the following sequence-to-function transformation: if

lim x→1−

− 1

log(1−x)

∞

k=0

1 k+1ukx

k+1

=A, (1.1)

thenuisL-summable toA. The matrix analogue of theL-summability method is the Ltmatrix [2] given by

ank= − 1 log1−tn

1 k+1t

k+1

n , (1.2)

where 0< tn<1 for allnand limntn=1. Thus, the sequenceuis transformed into the sequenceLtuwhosenth term is given by

Ltun= − 1 log1−tn

∞

k=0

1 k+1ukt

k+1

n . (1.3)

TheLtmatrix is called the logarithmic matrix. Throughout this paper,twill denote such a sequence: 0< tn<1 for alln, and limntn=1.

2. Basic notations and definitions. LetA=(ank)be an infinite matrix defining a sequence-to-sequence summability transformation given by

Axn=

∞

k=0

ankxk, (2.1)

this paper, we will use the following basic notations and definitions:

=

y:

∞

k=0

_ykis convergent

,

(A)= {y:Ay∈},

G=y:yk=Orkfor somer∈(0,1),

Gw=y:yk=Orkfor somer∈(0,w),0< w <1, Gw(A)=y:Ay∈Gw,

c= {the set of all convergent sequences}, c(A)=y:A(y)∈c.

(2.2)

Definition_2.1. _{If Xand Y are complex number sequences, then the matrixA is}

called anX-Ymatrix if the imageAuofuunder the transformationAis inYwhenever uis inX.

Definition2.2. The summability matrixA is said to be Gw-translative for a

se-quenceuinGw(A)provided that each of the sequencesTuandSuis inGw(A), where Tu= {u1,u2,u3,...}andSu= {0,u0,u1,...}.

Definition_2.3. _{The matrixAisGw-stronger than the matrixBprovided thatGw(B)} ⊆Gw(A).

3. Main results. Our first main result gives a necessary and sufficient condition for Ltto beGw-Gw.

Theorem3.1. The logarithmic matrixLtis aGw-Gwmatrix if and only if−1/log(1−

t)∈Gw.

Proof. Since 0< tn<1, it follows that

_ank

log1−tn≤ −1, (3.1)

for allnandk. Therefore, if−1/log(1−t)∈Gw, [3, Theorem 2.3] guarantees thatLt is aGw-Gw matrix. Conversely, if −1/log(1−t)∉Gw, then the first column ofLt is not inGw becausean,0= −tn/log(1−tn)∉Gw. Hence,Lt is not a Gw-Gw matrix by

[3, Theorem 2.3].

Corollary_3.2. _{If0< tn< un<1andLtis aGw-Gwmatrix, thenLuis also aGw-Gw}

matrix.

Corollary3.3. Supposeα >−1andLtis anGw-Gw matrix, then(1−t)α+1∈Gw.

Corollary3.4. Lettn=1−e−qn, whereqn=rn. ThenLtis aGw-Gw matrix if and

only ifr >1/w.

Gw

The next result suggests that the logarithmic matrixLtisGw-stronger than the iden-tity matrix. The result indicates that theLt matrix is rather a strong method in the Gw-Gw setting.

Theorem_3.6. _{IfLt is aGw-Gw matrix and the series} ∞

k=0xk has bounded partial sums, then it follows thatx∈Gw(Lt).

Proof. _{The proof easily follows using the same techniques as in the proof of}

Theorem 3.10 [2].

Remark3.7. Theorem 3.6indicates that ifLtis aGw-Gw matrix, thenGw(Lt)

con-tains the class of all conditionally convergent series. This suggests how large the size ofGw(Lt)is. In fact, we can give a further indication of the size ofGw(Lt)by showing that ifLtis aGw-Gwmatrix, thenGw(Lt)contains also an unbounded sequence. To see this, consider the sequencexgiven by

xk=(−1)k(k+1)2_{(k+2)(k+3). (3.2)}

Then

∞

k=0

1 k+1xkt

k+1

n =tn

∞

k=0

(−1)k(k+1)(k+2)(k+3)tnk= 6tn

1+tn4. (3.3)

Hence,

_Ltx n=

6tn

−log1−tn1+tn4≤ − 6

log1−tn. (3.4)

Thus, ifLtis anGw-Gwmatrix, then byTheorem 3.1,−1/log(1−t)∈Gw, sox∈Gw(Lt).

The next few results deal with theGw-translativity ofLt. We will show that theLt matrix isGw-translative for some sequences inGw(Lt).

Proposition_3.8. _{EveryGw-Gw Lt matrix isGw-translative for each sequencex}∈

Gw.

Theorem _3.9. _{SupposeLt is aGw-Gw matrix and} {xk/k}_{is a sequence such that}

xk/k=0for k=0, then the sequence {xk/k}is in Gw(Lt) for eachL-summable se-quencex.

Proof. _{LetY be theLt-transform of the sequence}{xk/k}_{. Then we have}

_{Yn= −} 1 log1−tn

∞

k=0

1 k(k+1)xkt

k+1

n

≤Cn+Dn, (3.5)

where

Cn= − x1tn

2 log1−tn−

_x2tn 6 log1−tn,

Dn= − 1 log1−tn

∞

k=3

1 k(k+1)xkt

k n .

ByTheorem 3.1, the hypothesis thatLtisGw-Gw implies thatC∈Gw, and hence there remains only to showD∈Gw to prove the theorem. Note that

Dn= −_log1 1−tn

∞

k=3

xk (k+1)

tn

0 t

k−1_dt

= − 1 log1−tn

tn

0 dt _∞

k=3

1 (k+1)xkt

k−1 .

(3.7)

The interchanging of the integral and the summation is legitimate as the radius of convergence of the power series

∞

k=3

1 k+1xkt

k−1 _(3.8)

is at least 1 by [2, Lemma 1] and hence the power series converges absolutely and uniformly for 0≤t≤tn. Now we let

F(t)=

∞

k=3

1 k+1xkt

k−1

. (3.9)

Then we have

−_logF(t) (1−t)= −

1 log(1−t)

∞

k=3

1 k+1xkt

k−1

, (3.10)

and the hypothesis thatx∈c(L)implies that

lim t→1−

F(t)

−log(1−t)=A (finite), for 0< t <1. (3.11)

We also have

lim t→0

F(t)

−log(1−t)=0. (3.12)

Now (3.11) and (3.12) yield

_−logF(t)_(1−t)≤M, for someM >0, (3.13)

and hence

Gw

So, we have

Dn= −_log1 1−tn

₀tnF(t)dt

≤ −_log1 1−tn

tn

0

_F(t)dt

≤ −_logM 1−tn

tn

0 −log(1−t)dt

= −M1−tn−_logMtn 1−tn

≤ −_logM 1−tn.

(3.15)

The hypothesis thatLtisGw-Gw implies that both−1/log(1−t)and(1−t)are inGw byTheorem 3.1. HenceD∈Gw.

Theorem_3.10. _{EveryGw-Gw Lt matrix isGw-translative for eachL-summable}

se-quence inGw(Lt).

Proof. Letx∈c(L)∩Gw(Lt). Then we will show that

(i) Tx∈Gw(Lt), (ii) Sx∈Gw(Lt).

We first show that (i) holds. Note that

_LtTx n= −

1 log1−tn

∞

k=0

1 k+1xk+1t

k+1

n

= −_log1 1−tn

∞ k=1 1 kxkt k n

= − 1 log1−tn

∞

k=1

₁

k+1+ 1 k(k+1)

xktk n ≤Pn+Qn,

(3.16)

where

Pn= −_log1 1−tn

∞

k=1

1 k+1xkt

k n ,

Qn= − 1 log1−tn

∞ k=1 1 k(k+1)xkt

k n . (3.17)

Next we will show that (ii) holds. We have

_LtSx n= −

1 log1−tn

∞

k=1

1 k+1xk−1t

k+1

n

= −_log1 1−tn

∞

k=0

1 k+2xkt

k+2

n ≤Wn+Un,

(3.18)

where

Wn= −_log11_−tn

∞

k=0

1 k+1xkt

k+2

n ,

Un= −_log1 1−tn

∞

k=0

1

(k+1)(k+2)xkt k+2

n .

(3.19)

The hypothesis thatX∈Gw(Lt)implies thatW∈Gw. We can also show thatU∈Gw by making a slight modification in the proof ofTheorem 3.9, replacing the sequence {xk/k}with the sequence{xk/(k+2)}. Hence, the theorem follows.

Acknowledgments. _{I would like to thank the referee for great assistance. I want to}

thank my wife Mrs. Tsehaye Dejene Habtemariam and my two great daughters Samera Mulatu and Abyssinia Mulatu for their great cooperation during my work on this paper. I also want to thank Dr. Harpal Sengh, the Chair of the Department of Natural Sciences and Mathematics, for his great support and encouragement.

References

[1] D. Borwein,A logarithmic method of summability, J. London Math. Soc.33(1958), 212–220. [2] M. Lemma,Logarithmic transformations intol1_{, Rocky Mountain J. Math.28(1998), no. 1,}

253–266.

[3] S. Selvaraj,Matrix summability of classes of geometric sequences, Rocky Mountain J. Math. 22(1992), no. 2, 719–732.

Mulatu Lemma: Department of Natural Sciences and Mathematics, Savannah State University, Savannah, GA 31406, USA