Let{pn}be a sequence of real or complex constants such that Pn=p0+p1

CHARACTERIZATION FOR RELATIONS ON SOME SUMMABILITY METHODS

W.T. Sulaiman

Abstract: In this paper we characterize a previous result proved by us connecting the summability methods|N , pn|k with either |N, qn|k or |N , wn|k for given sequences {pn},{qn}and{wn} of positive real constants. Other results are also deduced.

1 – Introduction

Let ^Pan be an infinite series with partial sums sn. Let σ_n^δ and η_n^δ denote the n-th Ces`aro mean of order δ (δ > −1) of the sequences {s_n} and {n a_n} respectively. The series^Pa_n is said to be summable |C, δ|_k,k≥1, if

∞

X

n=1

n^k−1|σ^δ_n−σ_n^δ₋₁|^k<∞ ,

or equivalently

∞

X

n=1

1

n|η^δ_n|^k<∞ .

Let{p_n}be a sequence of real or complex constants such that P_n=p0+p1+· · ·+p_n (p−1 =P−1 = 0). The series^Pa_n is said to be summable |N, p_n|if

(1.1)

∞

X

n=1

|T_n−T_n−1|<∞ ,

Received: October 19, 1997.

where

T_n= 1 Pn

n

X

v=0

p_n−vs_v (T−1 = 0). We writep={p_n} and

M =ⁿp: p_n>0 and p_n+1/p_n≤p_n+2/p_n+1≤1, n= 0,1, ...^o . It is known forp∈M (1.1) holds iff (see [5])

∞

X

n=1

1 n Pn

¯

¯

¯

¯

n

X

v=1

p_n−vv a_v

¯

¯

¯

¯

<∞ .

Forp∈M, we say that ^Pan is summable|N, pn|_k,k≥1, if

∞

X

n=1

1 n

¯

¯

¯

¯ 1 P_n

n

X

v=1

pn−vv av

¯

¯

¯

¯

k

<∞ (Sulaiman [6]).

In the special case in which p_n =A^r_n⁻¹, r > −1, where A^r_n is the coefficient of xⁿ in the power series expansion of (1−x)^−r−1 for|x|< 1, |N, p_n|_k reduces to

|C, r|_k summability. The series^Pa_n is said to be summable |N , p_n|_k,k≥1, if

∞

X

n=1

µP_n pn

¶k−1

|t_n−t_n−1|^k <∞ (Bor [1]), where

t_n= 1 P_n

n

X

v=0

p_vs_v .

If we takepn= 1, then|N , pn|_ksummability is equivalent to|C,1|_k summability.

In general these two summabilities are not comparable.

Throughout this paper we set

Q_n=q0+q1+· · ·+q_n, q−1 =Q−1 = 0 , Wn=w0+w1+· · ·+wn, w−1=W−1 = 0 ,

∆f_n=f_n−f_n+1 .

Let{pn}and{qn}be sequences of positive real constants such thatq ∈M. ^Pan

is said to be summable|N, p_n, q_n|_k,k≥1, if

∞

X

n=1

p_n P_nR_n^k₋₁

¯

¯

¯

¯

n

X

v=1

P_v−1q_n−va_v

¯

¯

¯

¯

k

<∞ (Sulaiman [7]) ,

where

Rn=p0qn+p1qn−1+· · ·+pnq0 . Clearly|N, p_n,1|_k is the same as |N , p_n|_k.

The following results are known.

Theorem A (Bor [1]). Let {p_n} be a sequence of positive real constants such that asn→ ∞

(1.2) (i) n p_n=O(P_n) ,

(ii) Pn=O(n pn) .

If^Pa_n is summable |C,1|_k, then it is summable|N , P_n|_k,k≥1.

Theorem B (Bor [2]). Let {p_n} be a sequence of positive real constants such that it satisfies (1.2). If^Pa_n is summable|N , p_n|_k then it is also summable

|C,1|_k.

Theorem C (Sulaiman [7]). Let {p_n}, {q_n} and {w_n} be sequences of positive real constants such that q ∈M and {p_n/P_nR^k_n−1} is nonincreasing for qn6=c. Lettndenote the(N , wn)-mean of the series^Pan. Let{εn}be a sequence of constants. If

m

X

n=v+1

p_nq_n−v−1

P_nR_n−1

=O(P_v⁻¹), m→ ∞ , (1.3)

∞

X

n=1

µP_n p_n

¶k−1

|ε_n|^k|∆t_n−1|^k <∞ , (1.4)

∞

X

n=1

p_n Pn

µW_n wn

¶k

|ε_n|^k|∆t_n−1|^k <∞ , (1.5)

∞

X

n=1

µP_n p_n

¶k−1µW_n w_n

¶k

|∆ε_n|^k|∆t_n−1|^k<∞ , (1.6)

and

∞

X

n=1

pn

P_n µPn−1

R_n−1

¶kµWn

w_n

¶k

|εn|^k|∆tn−1|^k<∞ , (1.7)

then the series^Pan is summable |N, pn, qn|_k,k≥1.

It may be mentioned that Theorems A and B are special cases of Theorem C.

The object of this paper is to prove the following

Theorem D. Let(p_n),(q_n)and(w_n)be sequences of positive real constants such thatq∈M and (pn/PnR^k_n−1) nonincreasing forqn6=c. Suppose that

R_n−1=O(P_n−1) , (1.8)

P_nw_n=O(p_nW_n) , (1.9)

∆

µw_nP_nR_n−1

W_np_nP_n−1

¶

=O µw_n

W_n

¶ , (1.10)

∆

µW_np_nP_n−1

w_nP_nR_n−1

εn

¶

=O

µp_nP_n−1

P_nR_n−1

¶ . (1.11)

Then necessary and sufficient conditions that ^Pa_nε_n be summable |N, p_n, q_n|_k whenever^Panis summable |N , wn|k,k≥1, are

(i) ε_n=O

µw_nP_nR_n−1

W_np_nP_n−1

¶ ,

(ii) ∆ε_n=O µ w_n

Wn−1

¶ .

2 – Lemmas

Lemma 1 (Sulaiman [7]). Let q∈M. Then for 0< γ≤1,

∞

X

n=v+1

q_n−v−1

n^γQ_n−1

=O(v^−γ).

Lemma 2 (Bor [4]). Letk≥1 and let A= (anv) be an infinite matrix. In order thatA∈(`<sup>k</sup>;`^k)it is necessary that

(2.1) a_nv =O(1) (for all n, v) .

Lemma 3. Suppose thatε_n =O(f_ng_n),f_n, g_n ≥0, f_n+1g_n+1 =O(f_ng_n),

∆(f_ng_n) =O(f_n) and ∆(ε_n/f_ng_n) =O(1/g_n). Then∆ε_n=O(f_n).

Proof: We have

ε_n=k_nf_ng_n, where k_n= ε_n

f_ng_n =O(1) ,

∆ε_n=k_nf_n+1∆g_n+k_ng_n+1∆f_n+f_n+1g_n+1∆k_n .

Since

f_n∆g_n+g_n+1∆f_n=O(f_n) , then

∆ε_n=k_nf_n∆g_n+k_n^hO(f_n)−f_n∆g_nⁱ+f_n+1g_n+1∆k_n

=k_nO(f_n) +O(f_ng_n|∆k_n|)

=O(f_n) +O(f_n)

=O(f_n) .

3 – Proof of Theorem D Write

Tn=

n

X

v=1

Pv−1qn−vavεv, tn= w_n W_nW_n−1

n

X

v=1

Wv−1av ,

(3.1) T_n=

n

X

v=1

W_v−1a_v

µP_v−1

Wv−1

q_n−vε_v

¶

=

n−1

X

v=1

µ v

X

r=1

Wr−1ar

¶

∆v

µPv−1

W_v−1

qn−vεv

¶ +

µ n

X

r=1

Wr−1ar

¶ Pn−1

W_n−1

q0εn

=

n−1

X

v=1

(

P_v−1∆_vq_n−v

W_v

w_v ε_vt_v+P_v−1q_n−v−1ε_vt_v +p_vq_n−v−1

Wv−1

w_v ε_vt_v−P_vq_n−v−1

Wv−1

w_v ∆ε_vt_v )

+P_n−1q0

Wn

w_n ε_nt_n

=T_n,1+T_n,2+T_n,3+T_n,4+T_n,5 , say.

In order to prove sufficiency, by Minkowski’s inequality, it is sufficient to show that

∞

X

n=1

p_n

P_nR^k_n−1 |T_n,r|^k<∞, r = 1,2,3,4,5.

Applying H¨older’s inequality,

m+1

X

n=2

p_n

P_nR^k_n−1|T_n,1|^k =

m+1

X

n=1

p_n P_nR^k_n−1

¯

¯

¯

¯

n−1

X

v=1

P_v−1∆_vq_n−v

W_v w_v ε_vt_v

¯

¯

¯

¯

k

≤

m+1

X

n=1

pn

PnR^k_n−1

n−1

X

v=1

P_v^k₋₁|∆_vq_n−v| µWv

w_v

¶k

|ε_v|^k|t_v|^k

·

½n−1

X

v=1

|∆vqn−v|

¾k−1

=O(1)

m

X

v=1

P_v^k₋₁ µW_v

wv

¶k

|ε_v|^k|t_v|^k

m+1

X

n=v

p_n|∆_vq_n−v| P_nR^k_n−1

=O(1)

m

X

v=1

p_v P_v

µP_v−1

R_v−1

¶kµW_v w_v

¶k

|εv|^k|tv|^k ,

m+1

X

n=2

p_n

P_nR^k_n−1|T_n,4|^k =

m+1

X

n=2

p_n P_nR^k_n−1

¯

¯

¯

¯

n−1

X

v=1

P_vq_n−v−1

W_v−1

w_v ∆ε_vt_v

¯

¯

¯

¯

k

≤

m+1

X

n=2

pn

P_nR_n−1 n−1

X

v=1

µPv

p_v

¶k

p_vq_n−v−1

µWv−1

w_v

¶k

|∆ε_v|^k|t_v|^k

·

½n−1

X

v=1

p_vq_n−v−1

R_n−1

¾k−1

=O(1)

m

X

v=1

µP_v pv

¶k

p_v

µW_v−1

wv

¶k

|∆ε_v|^k|t_v|^k

m+1

X

n=v+1

p_nq_n−v−1

PnRn−1

=O(1)

m

X

v=1

µPv

p_v

¶k−1µWv−1

w_v

¶k

|∆ε_v|^k|t_v|^k .

Similarly we can show that

m+1

X

n=2

p_n

P_nR^k_n−1 |T_n,2|^k=O(1)

m

X

v=1

µP_v p_v

¶k−1

|ε_v|^k|t_v|^k ,

m+1

X

n=2

p_n

P_nR^k_n−1 |T_n,3|^k=O(1)

m

X

v=1

p_v P_v

µW_v w_v

¶k

|ε_v|^k|t_v|^k ,

m

X

n=1

pn

PnR^k_n−1 |Tn,5|^k=O(1)

m

X

n=1

pn

P_n

µPn−1

R_n−1

¶kµWn

w_n

¶k

|εn|^k|tn|^k .

The sufficiency follows.

Necessity of (i). Using the result of Bor in [4], the transformation from ((Pn/pn)^1−1/ktn) into ([(pn/Pn)^1/k/Rn−1]Tn) maps `<sup>k</sup> into`^k and hence the di- agonal elements of this transformation are bounded (by Lemma 2) and so (i) is necessary.

Necessity of (ii). This follows from Lemma 3 and the necessity of (i) by taking f_n=w_n/W_n, g_n=P_nR_n−1/p_nP_n−1.

4 – Applications

Corollary 1. Let {p_n} and {w_n} be sequences of positive real constants such that (1.9) is satisfied.

Then the necessary and sufficient conditions such that ^Pan be summable

|N , p_n|_k whenever it is summable |N , w_n|_k,k≥1, is

(4.1) p_nW_n=O(P_nw_n) .

The proof follows from Theorem D by putting εn= 1,qn= 1.

Corollary 2 (Bor and Thorpe [3]). Let {p_n} and {w_n} be sequences of positive real constants such that (1.9) and (4.1) are satisfied.

Then the series^Pa_nis summable|N , p_n|_k iff it is summable |N , w_n|_k,k≥1.

The proof follows from Corollary 1.

Corollary 3. Let (p_n), (w_n) be sequences of positive real constants such that (1.9) is satisfied and

∆

µw_nP_n W_np_n

¶

=O µw_n

W_n

¶ ,

∆

µW_np_n w_nP_n ε_n

¶

=O µp_n

P_n

¶ .

Then necessary and sufficient conditions that ^Pa_nε_n be summable |N , p_n|_k whenever^Pa_nis summable |N , w_n|_k,k≥1, are

ε_n=O

µwnPn

W_np_n

¶

, ∆ε_n=O µwn

W_n

¶ .

The proof follows from Theorem D by putting q_n=ε_n= 1.

Remark. It may be mentioned that Theorems A and B could be obtained from Corollary 2.

REFERENCES

[1] Bor, H. – On two summability methods, Math. Proc. Cambridge Philos. Soc.,97 (1985), 147–149.

[2] Bor, H. –A note on two summability methods,Proc. Amer. Math. Soc.,98 (1986), 81–84.

[3] Bor, H. and Thorpe, B. – On some absolute summability methods,Analysis, 7 (1987), 145–152.

[4] Bor, H. – On the relative strength of two absolute summability methods, Proc.

Amer. Math. Soc., 113 (1991), 1009–1012.

[5] Das, G. – Tauberian theorems for absolute N¨orlund summability, Proc. London Math. Soc., 19 (1969), 357–384.

[6] Sulaiman, W.T. – Notes on two summability methods, Pure Appl. Math. Sci., 31(1990), 59–68.

[7] Sulaiman, W.T. – Relations on some summability methods, Proc. Amer. Math.

Soc., 118 (1993), 1139–1145.

W.T. Sulaiman, P.O. Box 120054, Doher – QATAR