On factor relations between weighted and Nörlund means

doi:10.5556/j.tkjm.50.2019.2704

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ON FACTOR RELATIONS BETWEEN WEIGHTED AND

NÖRLUND MEANS

G. CANAN HAZAR GÜLEÇ AND MEHMET ALI SARIGÖL

Abstract. By (X ,Y ), we denote the set of all sequences ǫ = (ǫn) such that Σǫnanis summable Y whenever Σan is summable X , where X and Y are two summability methods. In

this study, we get necessary and sufficient conditions for ǫ ∈¡¯

¯N , qn,un ¯ ¯_k, ¯ ¯ ¯N , pn ¯ ¯¢ and ǫ ∈¡¯ ¯ ¯N , p_n ¯ ¯, ¯ ¯N , q_n,u_n ¯

¯_k¢, k ≥ 1, using functional analytic tecniques, where ¯ ¯ ¯N , p_n ¯ ¯and ¯ ¯N , q_n,u_n ¯ ¯

kare absolute weighted and Nörlund summability methods, respectively, [1],

[5]. Thus, in the special case, some well known results are also deduced.

1. Introduction

Let A = (anv) be an infinite matrix of complex numbers, Σav be a given infinite series

with nth partial sum snand (un) be a sequence of nonnegative terms. Then the series Σavis

called summable |A,un|k, k ≥ 1, if (see [16]) ∞

X

n=0

uk−1_n |An(s) − An−1(s)|k< ∞, A−1(s) = 0, (1.1)

where A(s) = (An(s)) , the A-transform sequence of the sequence s = (sn), i.e., An(s) =

∞

X

ν=0 anνsν

converges for n ≥ 0. Note that if A is chosen as the Nörlund matrix (r e sp.un=n), then the

summability |A,un|k reduces to the absolute Nörlund summability

¯ ¯N , p_n,u_n ¯ ¯_k[5] (r e sp. the summability¯ ¯N , p_n ¯

¯_k, Borwein and Cass [2]), and also ¯ ¯N , p_n ¯ ¯₁= ¯ ¯N , p_n ¯ ¯, Mears [9]. Further, if pn=¡α+n−1_n ¢ and un=n, then the summability

¯

¯N , p_n,u_n ¯

¯_kis the same as the summability |C , α|kin Flett’s notation [4]. By a Nörlund matrix we mean one that

anv=

(

pn−v/Pn, 0 ≤ v ≤ n

0, v > n, (1.2)

Received December 27, 2017, accepted July 23, 2018.

2010 Mathematics Subject Classification. 40C05, 40D25, 40F05, 46A45.

Key words and phrases. Sequence spaces, absolute Nörlund summability, absolute weighted mean

summability, summability factors.

Corresponding author: G. Canan Hazar Güleç.

where¡pn¢ is a sequence of complex numbers with Pn=p0+p1+ · · · +pn6=0, P−(n+1)=0 for n ≥ 0. Also, if A =(anν) is the weighted matrix (r e sp.un=Pn/pn), i.e.

anν=

(

pv/Pn, 0 ≤ v ≤ n

0, v > n (1.3)

then the summability |A,un|kreduces to the summability

¯ ¯ ¯N , p_n,u_n ¯ ¯_k(r e sp. the summability ¯ ¯ ¯N , p_n ¯

¯_k, Bor [1]), where¡p_n¢ is a sequence of positive numbers such that P_n=p₀+p₁+ · · · +

pn→ ∞as n → ∞, Sulaiman [22]. For example, for the summability

¯ ¯ ¯N , p_n ¯ ¯_k, the condition (1.1) may be stated as ∞ X n=1 ¯ ¯ ¯ ¯ ¯ 1 Pn−1 µ pn Pn ¶1/k∗ ∞ X v=1 Pv−1av ¯ ¯ ¯ ¯ ¯ k < ∞.

Throughout this paper, k∗_{is the conjugate of k > 1, i.e., 1/k + 1/k}∗_{=1, and 1/k}∗_{=0 for} k = 1.

For any real α and integers n ≥ 0, we define

∆αǫ_n=

∞

X

v=n

A−α−1v−n ǫv

whenever the series on right side of equality is convergent.

Let ǫ be a sequence and X and Y be two methods of summability. If Σǫnanis summable Y whenever Σanis summable X , then ǫ is said to be a summability factor of type (X ,Y ) and

we denote it by ǫ ∈ (X ,Y ) [3] . The problems of summability factors dealing with absolute Cesàro and absolute weighted mean summabilities were widely examined by many authors (see [1-4],[8-11] ,[13-21]) e t al. For example, for α ≥ 0, k > 1, the summability factors of type ¡|C ,α| ,¯ ¯ ¯N , p_n ¯ ¯¢, ¡|C ,α|_k, ¯ ¯ ¯N , p_n ¯ ¯¢, (|C ,α|_k,|C ,1|) and¡|C ,1|_k, ¯ ¯ ¯N , p_n ¯ ¯¢ were characterized by Mohapatra [11], Mazhar [8], Mehdi [10], Sarıgöl and Bor [17] and Sarıgöl [18], respectively. In a more recent paper, Sarıgöl [13] has extended these classes to α > −1 and arbitrary positive sequence¡pn¢ in the following form.

Theorem 1.1. Let α > −1 and ¡pn¢ be arbitrary sequence of positive numbers. Then, necessary and sufficient condition for ǫ ∈¡|C ,α|k,

¯ ¯ ¯N , p_n ¯ ¯¢ ,k > 1, is ∞ X m=1 mαk∗+k∗−1 µ ∞ X n=m pn PnPn−1 ¯ ¯ ¯ ¯ ∞ X r =m A−α−1r −m ǫ_r r Pr −1 ¯ ¯ ¯ ¯ ¶k∗ < ∞. (1.4) 2. Main results

The purpose of this study is to generalize Theorem1.1by using Nörlund mean in place of Cesàro mean. Hence we characterize both classes¡¯

¯N , q_n,u_n ¯ ¯_k, ¯ ¯ ¯N , p_n ¯ ¯¢ and

¡¯ ¯ ¯N , p_n ¯ ¯, ¯ ¯N , q_n,u_n ¯

¯_k¢, which gives us more than we need. In the special cases, some well known results are also deduced.

Before stating the theorems we recall the following lemmas which plays important role for the proof our theorems.

Lemma 2.1. Let 1 < k < ∞. Then, A(x) ∈ ℓ whenever x ∈ ℓkif and only if ∞ X v=0 µ∞ X n=0 |anv| ¶k∗ < ∞, where ℓk=©x = (xν) : Σ|xν|k< ∞ª [13].

Lemma 2.2. Let 1 ≤ k < ∞. Then, A(x) ∈ ℓk whenever x ∈ ℓ if and only if

sup v ∞ X n=0 |anv|k< ∞, [7].

Now we begin with the theorem characterizing the class¡¯

¯N , q_n,u_n ¯ ¯_k, ¯ ¯ ¯N , p_n ¯ ¯¢ .

Theorem 2.3. Let q0be a non-zero number,(un) be a sequence of positive terms and (Cn) be a sequence satisfying n X v=0 Qn−vCv= ( 1, n = 0 0, n ≥ 1. (2.1)

Then necessary and sufficient condition for ǫ ∈¡¯

¯N , q_n,u_n ¯ ¯_k, ¯ ¯ ¯N , p_n ¯ ¯¢ , k > 1, is ∞ X m=1 1 um µ ∞ X n=m ¯ ¯ ¯ ¯ pn PnPn−1 n X r =m Pr −1ǫrGr m ¯ ¯ ¯ ¯ ¶k∗ < ∞ (2.2) where Gnr= n X v=r Cn−vQv. (2.3)

Proof. Let (tn) and (Tn) be the sequences of Nörlund mean ¡N , qn¢ and weighted mean

¡¯

N , pn¢ of the series Σanand Σǫnan, respectively, i.e tn= 1 Qn n X ν=0 qn−νsν= 1 Qn n X ν=0 Qn−νaν and Tn= 1 Pn n X ν=0 (Pn−Pν−1) ǫνaν.

Then we define sequences y =¡ yn¢ and ˜y = ¡ ˜yn¢ by yn=u1/k

∗

and ˜yn=Tn−Tn−1= pn PnPn−1 n X r =1 Pr −1ǫrar, n ≥ 1 and ˜y0=a0ǫ0 Then, ǫ ∈¡¯ ¯N , q_n,u_n ¯ ¯_k, ¯ ¯ ¯N , p_n ¯

¯¢ if and only if ˜y ∈ l whenever y ∈ l_k. On the other hand, since

q0is a non-zero, there exists a sequence (Cn) satisfying (2.1) and therefore it follows that tn= 1 Qn n X v=0 Qn−vav if and only if an= n X v=0 Cn−vQvtv.

Hence we get from (2.4),

an= n X v=0 Cn−vQv ν X r =0 u_r−1/k∗yr = n X r =0 u_r−1/k∗ n X v=r Cn−vQvyr= n X r =0 u−1/k_r ∗Gnryr

where Gnr is defined by (2.3), and so, for n ≥ 1,

˜yn= pn PnPn−1 n X r =1 Pr −1ǫrar = pn PnPn−1 n X r =1 Pr −1ǫr r X m=1 u_m−1/k∗Gr mym = pn PnPn−1 n X m=1 µ u_m−1/k∗ n X r =m Pr −1ǫrGr m ¶ ym = n X m=1 bnmym where bnm=      u−1/k_m ∗pn PnPn−1 Pn r =mPr −1ǫrGr m, m ≤ n 0, m > n. (2.5) Then, ˜y ∈ l whenever y ∈ lkif and only if

∞ X m=1 µ ∞ X n=m |bnm| ¶k∗ < ∞

by Lemma2.1, which is same as the condition (2.2). This completes the proof. _

It may be remarked that in the special case qn=Aα−1n and un=n, Theorem2.3reduces

to Theorem1.1. In fact, in this case it is obvious that¯

¯N , q_n,u_n ¯

¯_k= |C , α|_k. Also, we recall the following well known equality of Bosanquet and Das [3], for α 6= −1,−2,... , v ≥ 1,

n X r =v AαrA−α−2n−r = v Aα_vA−α−1_n−v n , (2.6)

Now, it is easy to see that C0=A−α−2₀ =1, Cn=A−α−2n and A α

−n=0 for n ≥ 1, and so, since

Gr m= r X ν=m AανA−α−2r −ν = m AαmA−α−1r −m r ,1 ≤ r ≤ v, and 0 for r > v,

by using (2.6), we get the matrix B = (bnm) as

bnm=      m1/kAα_mpn PnPn−1 n P r =mA −α−1 r −m ǫ_r r Pr −1, m ≤ n 0, m > n. So by Aα n∼n α

/Γ(α + 1) for α > −1 [4], it follows from applying Lemma2.1to the matrix B that (2.2) is the same as (1.4), as asserted.

Note that 1 ∈ (X ,Y ) leads us to a comparison of summability fields of methods X and

Y , where 1 =(1,1,...) . So taking ǫn=un=1 for all n ≥ 1 in Theorem2.3we get the following

result. Corollary 2.4. If k > 1, then, 1 ∈¡¯ ¯N , q_n ¯ ¯_k, ¯ ¯ ¯N , p_n ¯ ¯¢ if and only if ∞ X m=1 µ ∞ X n=m ¯ ¯ ¯ ¯ pn PnPn−1 n X r =m Pr −1Gr m ¯ ¯ ¯ ¯ ¶k∗ < ∞. (2.7)

This result also extends the following result of Kayashima [6] to k > 1.

Corollary 2.5. If¡pn¢ and ¡qn¢ are positive and nonincreasing sequences and ¡qn+1/qn¢ is nondecreasing, then 1 ∈¡¯ ¯N , q_n ¯ ¯, ¯ ¯ ¯N , p_n ¯ ¯¢ .

Theorem 2.6. Let k ≥ 1 and (un) be a sequence of nonnegative terms. Then, ǫ ∈¡¯ ¯ ¯N , p_n ¯ ¯, ¯ ¯N , q_n,u_n ¯ ¯_k¢ if and only if sup ν ½ u1/kν ∗ ¯ ¯ ¯ ¯ ǫ_νPv Qvpν ¯ ¯ ¯ ¯ ¾ < ∞ (2.8) and sup ν ∞ X n=v+1 ¯ ¯ ¯ ¯ u1/k_n ∗ µ Ωnνq ǫν Pν pν− Ω q n,ν+1ǫν+1 Pν−1 pν ¶¯ ¯ ¯ ¯ k < ∞, (2.9) where Ωqnν= Qn−ν Qn −Qn−ν−1 Qn−1 ,Q−1=0. (2.10)

Proof. As in proof of Theorem2.3, we define sequences y =¡ yn¢ and ˜y = ¡ ˜yn¢ by y0= ǫ0a0, yn=u1/k ∗ n (tn−tn−1) = u1/k ∗ n n X ν=0 µ Qn−ν Qn −Qn−ν−1 Qn−1 ¶ ǫ_{νaν,n ≥ 1 (2.11)}

˜y0=a0, ˜yn=Tn−Tn−1= pn PnPn−1 n X r =1 Pr −1ar, n ≥ 1 (2.12) Then, ǫ ∈¡¯ ¯ ¯N , p_n ¯ ¯, ¯ ¯N , q_n,u_n ¯

¯_k¢ iff y ∈ l_k whenever ˜y ∈ l . On the other hand, from (2.12) we write an= Pn pn ˜yn− Pn−2 pn−1

˜yn−1,n ≥ 1 and a0=˜y0

Hence, by (2.11) we get yn=u1/k ∗ n n X ν=1Ω q nνǫνaν=u1/k ∗ n n X ν=1Ω q nνǫν µ Pν pν ˜yν− Pν−2 pν−1 ˜yν−1 ¶ =u1/k_n ∗ ( Ωqnnǫn Pn pn ˜yn+ n−1 X ν=1 µ Ωqnνǫν Pν pν− Ω q n,ν+1ǫν+1 Pν−1 pν ¶ ˜yν ) = n X ν=1 cnν˜yν where cnν=        u_n1/k∗³ΩqnνǫνP_pν ν− Ω q n,ν+1ǫν+1 Pν−1 pν ´ , 1 ≤ ν ≤ n − 1 u_n1/k∗ΩqnnǫnP_pn_n, ν =n 0, ν >n.

So y ∈ lkwhenever ˜y ∈ l if and only if

sup ν ∞ X n=v |cnν|k< ∞

by Lemma2.2or, equivalently, sup ν ½ u1/kν ∗ ¯ ¯ ¯ ¯ ǫ_νPv Qvpν ¯ ¯ ¯ ¯ ¾ < ∞, ν ≥ 1 and sup ν ∞ X n=v+1 ¯ ¯ ¯ ¯u 1/k∗ n µ Ωqnνǫν Pν pν− Ω q n,ν+1ǫν+1 Pν−1 pν ¶¯ ¯ ¯ ¯ k < ∞.

Thus the proof is completed. 

Corollary 2.7. If k ≥ 1, then, 1 ∈ ¡¯ ¯ ¯N , p_n ¯ ¯, ¯ ¯N , q_n ¯ ¯_k¢ if and only if sup ν ½ v1/k∗ ¯ ¯ ¯ ¯ Pv Qvpν ¯ ¯ ¯ ¯ ¾ < ∞ (2.13) and sup ν ∞ X n=v+1 ¯ ¯ ¯ ¯ n1/k∗ µ Ωqnν Pν pν− Ω q n,ν+1 Pν−1 pν ¶¯ ¯ ¯ ¯ k < ∞. (2.14)

Proof. Put ǫn=1 and un=n for all n ≥ 1 in Theorem2.6.

This result, for k = 1, reduces to the following theorem of Kayashima [6].

Corollary 2.8. If ¡pn¢ and ¡qn¢ are positive and nondecreasing sequences and ¡qn+1/qn¢ is nonincreasing, then 1 ∈¡¯

¯ ¯N , pn¯¯,¯¯N , qn¯¯¢ .

Proof. By considering that¡pn¢ and ¡qn¢ are positive and nondecreasing sequences, we have Pn

Qnpn

≤n + 1

Qn

≤q−1₀ for all n ≥ 0.

Also, by hypotheses on the sequence¡qn¢, it converges to a number, limnqn+1/qn= σsay. So,

there exists a nonincreasing null sequence (xn) such that qn+1=(σ+xn)qnfor all n ≥ 0, where σ ≥1. Then, it can be written that

Qn=q0+ σQn−1+ n X v=1 qv−1xv−1 which gives Qn Qn−1 = q0 Qn−1 + σ +Zn→ σas n → ∞ (2.15) where Zn= 1 Qn−1 n X v=1 qv−1xv−1

Since (xn) is nonincreasing, it is easily seen that (Zn) is nonincreasing, which implies that

(Qn/Qn−1) is nonincreasing. So it follows that, for 0 ≤ v ≤ n,

Ωnνq = Qn−v−1 Qn µ _Q n−v Qn−v−1 − Qn Qn−1 ¶ ≥0 Further, Cnv= Ωqnν Pν pν− Ω q n,ν+1 Pν−1 pν ≥0.

In fact, if qn−v/Qn−qn−v−1/Qn−1≥0, then it is clear that Cnv≥0, since Cnv= Pν pν µ qn−v Qn −qn−v−1 Qn−1 ¶ + Ωq_n,ν+1. (2.16)

If qn−v/Qn−qn−v−1/Qn−1<0, then, it can be deduced from the condition on¡qn¢ that qn−v Qn −qn−v−1 Qn−1 ≥ qn−v−1 Qn µ _q n−m qn−m−1 − Qn Qn−1 ¶ ≥ qn−m Qn −qn−m−1 Qn−1 , 0 ≤ m ≤ v,

which implies vµ qn−v Qn −qn−v−1 Qn−1 ¶ ≥ v−1 X m=0 µ qn−m Qn −qn−m−1 Qn−1 ¶ = −Ωqnv.

Also, since ¡pv¢ is a positive nondecreasing sequence, we can write Pv ≤v pv for all v ≥ 1,

which gives us, by (2.17),

Pν pν µ qn−v Qn −qn−v−1 Qn−1 ¶ ≥ − Pν v pνΩ q nv≥ −Ω q nv.

This means that Cnv≥0 for 0 ≤ v ≤ n. Hence, by considering (2.15), we have

sup v ∞ X n=v+1 ¯ ¯ ¯ ¯ Pν pνΩ q nν− Pν−1 pν Ω q n,ν+1 ¯ ¯ ¯ ¯ =sup v lim m m X n=v+1 µ Pν pνΩ q nν− Pν−1 pν Ω q n,ν+1 ¶ ≤sup v limm · Pν pν µ Qm−v Qm − q0 Qv ¶ −Pv−1Qm−ν−1 pvQm ¸ =sup v · Pν pν µ ₁ σv − q0 Qv ¶ − Pv−1 pvσv−1 ¸ < ∞,

which completes the proof. _

Further the following result of [12] is obtained form Corollary2.7by choosing qn=1 for n ≥ 1. Corollary 2.9. If k ≥ 1, then, 1 ∈ ¡¯ ¯ ¯N , pn¯¯,|C ,1|_k¢ if and only if sup ν Pv v1/kpν < ∞. References

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Department of Mathematics, Pamukkale University, TR-20007 Denizli, Turkey. E-mail:gchazar@pau.edu.tr

Department of Mathematics, Pamukkale University, TR-20007 Denizli, Turkey. E-mail:msarigol@pau.edu.tr