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A GENERALIZATION THEOREM COVERING MANY
ABSOLUTE SUMMABILITY METHODS
W. T. SULAIMAN
(Received 27 April 1999)
Abstract.A general theorem concerning many absolute summability methods is proved. 2000 Mathematics Subject Classification. 40F05, 40G05, 40D15, 40D25.
1. Introduction. Letanbe given infinite series with the sequence of partial sums (sn). Byσnδandµnδwe denotenth Cesaro mean of orderδ (δ >−1)of the sequences (sn)and(nan), respectively. The seriesanis said to be summable|C,δ|k,k≥1 if
∞
n=1
nk−1σδ
n−σnδ−1k<∞, (1.1)
or equivalently
∞
n=1 n−1µδ
nk<∞. (1.2)
Let(pn)be a sequence of real or complex numbers such that
Pn= n
ν=0
pν → ∞ asn→ ∞P−i=p−i=0, i≥1. (1.3)
The sequence-to-sequence transformation:
tn=P1 n
n
ν=0
pνsν, (1.4)
defines the sequence(tn)of the Riesz means of the sequence(sn)generated by the
sequence of coefficients(pn)(see [4]). The seriesanis said to be summable|R,pn|k, k≥1, if
∞
n=1 nk−1t
n−tn−1k<∞. (1.5)
The seriesanis said to be summable|N,p¯ n|k,k≥1 (see [1]), if
∞
n=1 P
n pn
k−1
tn−tn−1k<∞. (1.6)
Forpn=1/(n+1), the summability|N,p¯ n|kis equivalent to|R,logn,1|k. The series
anis said to be summable|N,pn|, if
∞
n=1
Tn−Tn−1<∞, (1.7)
where
Tn=P1 n
n
ν=0
pn−νsν T−1=0. (1.8)
We writep= {pn}and
M=
p:pn>0 and ppn+1 n ≤
pn+2
pn+1≤1, n=0,1,... . (1.9)
It is known that forp∈M, equation (1.7) holds if and only if (see [3])
∞
n=1
1 nPn
n
ν=1
pn−ννaν
<∞. (1.10)
Forp∈M, the seriesanis said to be summable|N,pn|k,k≥1 (see [5]), if
∞
n=1
1 nPk
n
n
ν=1
pn−ννaν k
<∞. (1.11)
In the special case in whichpn=Arn−1, r >−1, whereArnis the coefficient ofxnin
the power series of(1−x)−r−1, for|x|<1,|N,pn|ksummability reduces to |C,r|k
summability, and forpn=1/(n+1),|N,pn|kis equivalent to|N,1/(n+1)|k
summa-bility. We assume that(fn),(gn),(Gn), and(Hn)are positive sequences.
2. Main result. We prove the following theorem.
Theorem2.1. Let
Yn=F 1 n−1Hn
n
ν=1
νfn−νxν"ν (2.1)
such that
Fn= n
ν=1
fν → ∞, f∈M, Xn=G1 n
n
ν=1
gνxν. (2.2)
Suppose that{nGn"n/FngnHn}is monotonic and
gn+1=Ogn, (2.3)
∆gn=O g
n+1 n
, (2.4)
g
nHn nGn
=O
g
nHn nGnFn+1
, (2.5)
m
n=ν+1 fn−ν Fn−1Hnk=O
1
Hk ν
Then necessary and sufficient conditions that|Yn|k<∞whenever|Xn|k<∞are
(i) "n=O(FngnHn/nGn),
(ii) ∆"n=O(gn+1Hn/nGn).
3. Lemmas
Lemma3.1(see [2]). Letk≥1, and letA=(anν)be an infinite matrix. In order that A∈(&k;&k)it is necessary that
anν=O(1) (alln,ν). (3.1)
Lemma3.2(see [6]). Letp∈M. Then for0< r≤1,
∞
n=ν+1 pn−ν−1 nrPn−1=O
ν−r. (3.2)
Lemma3.3. Suppose that"n=O(fngn),fn,gn≥0,{"n/fngn}monotonic,gn= O(1), andfn=O(fn/gn+1). Then"n=O(fn).
Proof. Letkn="n/fngn=O(1). If(kn)is nondecreasing, then
"n=knfngn−kn+1fn+1gn+1≤knfngn−knfn+1gn+1 =knfngn=knfngn+gn+1fn,
"n=O
fngn+Ogn+1fn=Ofn+Ofn=Ofn.
(3.3)
If(kn)is nonincreasing, write∇fn=fn+1−fn,
∇"n=kn+1fn+1gn+1−knfngn≤kn∇fngn =knfn∇gn+gn+1∇fn,
"n=∇"n=O
fn∇gn+Ogn+1∇fn =Ofn∇gn+Ogn+1fn
=Ofn+Ofn=Ofn.
(3.4)
4. Proof of Theorem2.1
Sufficiency. We have via Abel’s transformation:
Yn=F 1 n−1Hn
n
ν=1 gνxν
νfgn−ν
ν "ν
=F 1 n−1Hn
n−1
ν=1 ν
r=1 grxr
νfn−ν gν "ν
+
n
r=1 grxr
nf0
gn"n
=F 1 n−1Hn
n−1
ν=1 GνXν
−fgn−ν
ν "ν+(ν+1)g −1
ν fn−ν"ν+(ν+1)gν−+11νfn−ν"ν
+(ν+1)g−1
ν+1fn−ν−1"ν +FnGnXnf0 n−1Hngn"n =Yn,1+Yn,2+Yn,3+Yn,4+Yn,5.
By Minkowski’s inequality,
m
n=1
Ynk=O(1)m n=1
5
r=1
Yn,rk. (4.2)
Applying Hölder’s inequality,
m+1
n=2
Yn,1k=m+1 n=2
1 Fk
n−1Hnk n−1
ν=1 fn−νGgν
νXν"ν k
≤m+1 n=2
1 Fn−1Hnk
n−1
ν=1 fn−ν
G ν gν
k
Xνk"νk
n−1
ν=1 fn−ν Fn−1
k−1
≤O(1) m
ν=1 G
ν gν
k
Xνk"νk m+1
n=ν+1 fn−ν Fn−1Hnk
≤O(1) m
ν=1
1 Hk
ν
ν
Fν−1 kG
ν gν
k
Xνk"νk,
m+1
n=2
Yn,2k=m+1 n=2
1 Fk
n−1Hnk
n−1
ν=1
(ν+1)Gνg−1
ν fn−νXν"ν k
≤O(1) m+1
n=2
1 Fn−1Hnk
n−1
ν=1 νkGk
νgν−1kfn−νXνk"νk
n−1
ν=1 fn−ν Fn−1
k−1
≤O(1) m
ν=1 νkGk
νgν−1kXνk"ν m+1
n=ν+1 fn−ν Fn−1Hnk
=O(1) m
ν=1 ν
Hν k
Gk ν
gνk gk
νgkν+1
Xνk" νk
≤O(1) m
ν=1
1 Hνk
ν
Fν−1 kG
ν gν
k
Xνk"νk,
m+1
n=2
Yn,3k=m+1 n=2
1 Fk
n−1Hnk
n−1
ν=1 νg−1
ν+1Gννfn−νXν"ν k
≤m+1 n=2
1 Fk
n−1Hnk n−1
ν=1 νk Gν
gν+1 k
νfn−νXνk"νk
n−1
ν=1
fn−ν
k−1
≤O(1) m
ν−1 νkGν
gν k
Xνk"νk m+1
n=ν+1 νfn−ν
Fk n−1Hnk
≤O(1) m
ν=1
1 Hk
ν
ν
Fν−1 kG
ν gν
k
m+1
n=2
Yn,4k=m+1 n=2
1 Fk
n−1Hnk
n−1
ν=1 νg−1
ν+1fn−ν−1GνXν"ν k
≤ m+1
n=2
1 Fn−1Hnk
n−1
ν=1 νk Gν
gν+1 k
fn−ν−1Xνk"νk
n−1
ν=1 fn−ν−1
Fn−1
k−1
≤O(1) m
ν=1 νk Gν
gν+1 k
Xνk"νk m+1
n=ν+1 fn−ν−1 Fn−1Hnk
≤O(1) m
ν=1 ν
Hν k G
ν gν+1
k
Xνk"νk,
m
n=1
|Yn,5|k= m n=1
nGnXnf0"n Fn−1Hngn
k
≤O(1) m
n=1
n
Fn−1 kG
n gn
k 1
Hnk
Xnk" nk.
(4.3)
Sufficiency of (i) and (ii) follows. Necessity of (i): using the result of Bor in [2], the transformation from(Xn)into(Yn)maps&kinto&kand hence the diagonal elements
of this transformation are bounded (byLemma 3.1) and so (i) is necessary. Necessity of (ii): this follows fromLemma 3.3and necessity of (i) by taking
fn≡gnGnHn
n , gn≡Fnusing (2.3). (4.4)
This completes the proof of the theorem.
Remarks. It may be mentioned that on putting
(1)fn=pnandHn=n1/k, we obtain|N,pn|ksummability ofan"n.
(2)gn=Qn−1andGn=Qn−1(Qn/qn)1/k, we obtain|N,q¯ n|ksummability ofan.
(3)gn=Qn−1andGn=n1/k−1(QnQn−1/qn), we obtain|R,qn|ksummability ofan.
5. Applications
Theorem5.1. Letp∈M,{(n"n/Pn)(Qn/nqn)1/k}is monotonic,
1 n
nq n Qn
1/k =O
1 n
nq n Qn
1/k 1 Pn+1
, nqn=OQn. (5.1)
Then necessary and sufficient conditions thatan"nbe summable|N,pn|kwhenever
anis summable|N,q¯ n|k,k≥1, are
"n=O
Pn n
nq n Qn
1/k
, "n=O
1 n
nq n Qn
1/k
. (5.2)
Theorem5.2. Letp∈M,{Qn"n/Pnqn}is monotonic,
qn Qn
=O qn QnPn+1
Then necessary and sufficient conditions thatan"nbe summable|N,pn|k
when-everanis summable|R,qn|k,k≥1 are
"n=O P
nqn Qn
, "n=O q
n Qn
. (5.4)
Corollary 5.3. Necessary and sufficient conditions that an"n be summable |C,α|K,0< α≤1wheneveranis summable|C,1|k,k≥1, are
"n=Onα−1, "n=On−1, (5.5)
provided{n1−α"n}is monotonic.
Corollary 5.4. Necessary and sufficient conditions that an"n be summable |N,1/(n+1)|kwheneveranis summable|C,1|k,k≥1, are
"n=O logn
n
, "n=On−1, (5.6)
provided{n"n/logn}is monotonic.
Corollary 5.5. Necessary and sufficient conditions that an"n be summable |N,1/(n+1)|kwheneveranis summable|R,logn,1|k,k≥1, are
"n=O
(logn)1−1/k n
, "n=O
1
n(logn)1/k , (5.7)
provided{n(logn)1/k−1"n}is monotonic.
Corollary 5.6. Necessary and sufficient conditions that an"n be summable |C,α|k,0< α≤1wheneveranis summable|R,logn,1|k,k≥1, are:
"n=O
nα−1
(logn)1/k , "n=O
1
n(logn)1/k , (5.8)
provided{n1−α(logn)1/k"n}is monotonic.
References
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[3] G. Das,Tauberian theorems for absolute Nörlund summability, Proc. London Math. Soc. (3)
19(1969), 357–384.MR 39#1850. Zbl 183.32603.
[4] G. H. Hardy, Divergent Series, Oxford University Press, Oxford, 1949. MR 11,25a. Zbl 032.05801.
[5] W. T. Sulaiman,Notes on two summability methods, Pure Appl. Math. Sci.31(1990), no. 1-2, 59–69.MR 91m:40010. Zbl 713.40003.
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