On |A|k summability factors of infinite series

doi: 10.3176/proc.2010.3.02 Available online at www.eap.ee/proceedings

On |A|

summability factors of infinite series

B. E. Rhoades^aand Ekrem Savas¸^b∗

a Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, U.S.A.; [email protected]

b Department of Mathematics, Istanbul Commerce University, Uskudar, Istanbul, Turkey

Received 18 February 2009, revised 1 September 2009, accepted 8 September 2009

Abstract. In an earlier paper (Rhoades, B. E. and Savas¸, E. Some necessary conditions for absolute matrix summability factors.

Indian J. Pure Appl. Math., 2002, 33(7), 1003–1009) the authors obtained necessary conditions for the series ∑anto be absolutely summable of order k by a triangular matrix. In this paper we present sufficient conditions for absolute matrix summability factors.

As a corollary we obtain a result of N. Singh (On |N, pn| summability factors of infinite series. Indian J. Math., 1968, 10, 19–24).

Key words: absolute summability, summability factors.

Let A be a lower triangular matrix, {s_n} any sequence. Then

A_n:=

∑

a_nνs_ν.

A series ∑an, with partial sums s_n, is said to be summable |A|_k, k ≥ 1 if

∑

n^k−1|A_n− A_n−1|^k< ∞.

We may associate with A two lower triangular matrices A and ˆA as follows:

¯a_nν=

∑

a_nr, n,ν = 0, 1, 2, . . . ,

and

ˆa_nν = ¯a_nν− ¯a_n−1,ν, n = 1, 2, 3, . . . .

In our previous work on absolute summability [1,2] we have assumed that the triangular matrix A had row sums one. This condition rules out the consideration of factorable matrices that are not weighted mean matrices. A lower triangular matrix A is said to be factorable if the nonzero terms a_nkcan be written as a_nb_k for 0 ≤ k ≤ n. If A is a factorable matrix with row sums one, then it is a weighted mean matrix.

∗ Corresponding author, [email protected]

We shall first establish a general theorem for triangular matrices, which also applies to factorable matrices which need not be weighted mean matrices, and then we shall specialize this result to triangular matrices with row sums one.

A series ∑anwith partial sums s_nis said to be bounded |A|_k, k ≥ 1, if ∑^m_ν=1a_mν|s_ν|^k= O(1) as m → ∞.

Theorem 1. Let A be a lower triangular matrix satisfying (i)

n−1

∑

ν=1

|∆_νˆa_nν| = O(|a_nn|), n ≥ 2,

(ii)

m+1

∑

n=ν+1

|∆_νˆa_nν| = O(|a_νν|), m ≥ν, (iii) n|a_nn| = O(1),

(iv) |a_νr− a_ν+1,r| = O(|a_ν+1,ν+1a_νr|), 0 ≤ r ≤ν,

(v) ⁿ⁻¹

∑

ν=1

|a_ννˆa_n,ν+1| = O(|a_nn|), n ≥ 2, and

(vi)

m+1

∑

n=ν+1

| ˆa_n,ν+1| = O(1), m ≥ν... .

If ∑anis bounded |A|_kand {λ_n} is a bounded nonzero sequence satisfying (vii)

∑

|a_nn||λ_n|^k= O(1), and (viii) |∆|λ_n|^k| = O(|a_nn||λ_n|^k),

then the series ∑anλ_nis summable |A|_k, k > 1.

Proof. Let (y_n) be the nth term of the A-transform of ∑ⁿ_i=0λ_ia_i. Then

y_n=

∑

a_nis_i=

∑

a_ni

∑

λ_νa_ν

=

∑

λ_νa_ν

∑

a_ni=

∑

¯a_nνλ_νa_ν

and, for n > 0,

Y_n:= y_n− y_n−1=

∑

( ¯a_nν− ¯a_n−1,ν)λ_νa_ν =

∑

ˆa_nνλ_νa_ν.

Using Abel’s transformation, we have, for n > 1,

Y_n:=

n−1

∑

ν=1

(∆_νˆa_nν)λ_νs_ν+

n−1

∑

ν=1

ˆa_n,ν+1(∆λ_ν)s_ν+ a_nnλ_ns_n

= T_n1+ T_n2+ T_n3, say.

Since Y₁is bounded, in order to prove our theorem, it is sufficient, by Minkowski’s inequality, to show that

∑

n^k−1|T_nr|^k< ∞, for r = 1, 2, 3.

Using H¨older’s inequality and (i), (iii), and (ii),

I₁: =

m+1

∑

n=2

n^k−1|T_n1|^k=

∑

n^k−1

¯¯

¯

n−1

∑

ν=1

∆_νˆa_nνλ_νs_ν

¯¯

¯^k

= O(1)

m+1

∑

n=2

n^k−1

³n−1 ν

∑

|∆_νˆa_nν||λ_ν||s_ν|

´_k

= O(1)^m+1

∑

n=2

n^k−1

³n−1 ν

∑

|∆_νˆa_nν||λ_ν|^k|s_ν|^k

´

×

³n−1 ν

∑

|∆_νˆa_nν|

´_k−1

= O(1)

m+1

∑

n=2

(n|a_nn|^k−1

n−1

∑

ν=1

|∆_νˆa_n,ν||λ_ν|^k|s_ν|^k

= O(1)

∑

ν=1

|λ_ν|^k|s_ν|^{k m+1}

∑

n=ν+1

|∆_νˆa_nν|

= O(1)

∑

|a_νν||λ_ν|^k|s_ν|^k.

Using the boundedness of ∑anand {λ_n}, (iv), (viii), and (vii),

I₁= O(1)

∑

|λ_ν|^k h ν

i=0

∑

|a_νi||s_i|^k−

ν−1 i=0

∑

|a_νi||s_i|^k i

= O(1) h m

ν

∑

|λ_ν|^k

∑

|a_νi||s_i|^k−^m−1

∑

ν=0

|λ_ν+1|^k

∑

|a_ν+1,i||s_i|^k i

≤ O(1) h

|λ_m|^k

∑

|a_mi||s_i|^k

+

m−1

∑

ν=1

³

|λ_ν|^k

∑

|a_νi||s_i|^k− |λ_ν+1|^k

∑

|a_ν+1,i||s_i|^k

´i

≤ O(1) + O(1) hm−1

ν

∑

³

|λ_ν|^k− |λ_ν+1|^k

´ ν i=0

∑

|a_νi||s_i|^k

+

m−1

∑

ν=1

|λ_ν+1|^k

∑

|a_νi− a_ν+1,i||s_i|^k i

= O(1) + O(1)

m−1

∑

ν=1

|∆(|λ_ν|^k)| + O(1)

m−1

∑

ν=1

|λ_ν+1|^k|a_ν+1,ν+1|

∑

|a_νi||s_i|^k

= O(1) + O(1)

m−1

∑

ν=1

|a_νν||λ_ν|^k+ O(1)

m−1

∑

ν=1

|a_ν+1,ν+1||λ_ν+1|^k

= O(1).

Using (viii), H¨older’s inequality, (v), (iii), and (vi),

I₂: =

m+1

∑

n=2

n^k−1|T_n2|^k=

m+1

∑

n=2

n^k−1

¯¯

¯

n−1

∑

ν=1

ˆa_n,ν+1(∆λ_ν)s_ν

¯¯

¯^k

≤^m+1

∑

n=2

n^k−1 hn−1

ν

∑

| ˆa_n,ν+1||∆λ_ν||s_ν| i_k

= O(1)

m+1

∑

n=2

n^k−1 hn−1

ν

∑

|λ_ν|^k|s_ν|^k|a_ννˆa_n,ν+1| i

× hn−1

ν

∑

|a_ννˆa_n,ν+1| i_k−1

= O(1)

m+1

∑

n=2

(n|a_nn|)^k−1

n−1

∑

ν=1

| ˆa_n,ν+1a_νν||λ_ν|^k|s_ν|^k

= O(1)

∑

ν=1

a_νν|λ_ν|^k|s_ν|^{k m+1}

∑

n=i+1

| ˆa_n,ν+1|

= O(1)

∑

a_νν|λ_ν|^k|s_ν|^k= O(1),

as in the proof of I₁. Finally, using (iii),

∑

n^k−1|T_n3|^k=

∑

n^k−1

¯¯

¯a_nnλ_ns_n

¯¯

¯^k

= O(1)

∑

n=1

n^k−1|a_nn|^k|λ_n|^k|s_n|^k

= O(1)

∑

(n|a_nn|)^k−1a_nn|λ_n|^k−1|λ_n||s_n|^k

= O(1)

∑

|a_nnλ_n||s_n|^k

= O(1), as in the proof of I₁.

Theorem 2. Let A be a lower triangular matrix with nonnegative entries satisfying (ix) ¯a_n0= 1, n = 0, 1, 2, . . .,

(x) a_n−1,ν ≥ a_nν for n ≥ν + 1, and conditions (iii)–(v) of Theorem 1.

If ∑a_nis bounded |A|_kand {λ_n} is a bounded nonzero sequence satisfying conditions (vii) and (viii) of Theorem 1, then the series ∑anλ_nis summable |A|_k, k > 1.

Proof. Upon examining the conditions of Theorem 1 it is clear that one needs to show that conditions (ix) and (x) imply that ˆa_n,ν+1≥ 0 and that conditions (i), (ii), and (vi) of Theorem 1 hold.

Using the definitions of ˆa_nν and ¯a_nν, and (ix) and (x),

∆_νˆa_nν= ˆa_nν− ˆa_n,ν+1

= ¯a_nν− ¯a_n−1,ν− ¯a_n,ν+1+ ¯a_n−1,ν+1

= a_nν− a_n−1,ν≤ 0.

Therefore

n−1

∑

ν=1

|∆_νˆa_nν| =ⁿ⁻¹

∑

ν=1

(a_n−1,ν− a_nν)

= 1 − 1 + a_n0+ a_nn≤ a_nn, and condition (i) of Theorem 1 is true.

Also,

m+1

∑

n=ν+1

|∆_νˆa_nν| =

m+1

∑

n=ν+1

(a_n−1,ν− a_νν) = a_νν− a_m+1,ν≤ a_νν, and condition (ii) of Theorem 1 is true.

Finally,

m+1

∑

n=ν+1

| ˆa_n,ν+1| =

m+1

∑

n=ν+1

∑

(a_n−1,i− a_ni)

=

∑

m+1

∑

n=ν+1

(a_n−1,i− a_ni)

=

∑

(a_νi− a_m+1,i) ≤

∑

a_νi= 1,

and condition (vi) of Theorem 1 is satisfied.

If one is dealing with absolute summability of order 1, then conditions (iii) and (iv) of Theorem 1 are not needed.

Theorem 3. Let A be a lower triangular matrix satisfying conditions (ii), (iv), and (vi) of Theorem 1. If ∑a_n is bounded |A| and {λ_n} is a bounded nonzero sequence satisfying conditions (vii) and (viii) of Theorem 1 (with k = 1), then the series ∑anλ_nis summable |A|.

Proof. This can be proved by using the techniques similar to that of Theorem 1. So we omit it.

Theorem 4. Let A be a lower triangular matrix with nonnegative entries satisfying conditions (ix) and (x) of Theorem 2 and condition (iv) of Theorem 1. If ∑anis bounded |A| and {λ_n} is a bounded nonzero sequence satisfying conditions (vii) and (viii) of Theorem 1, then the the series ∑anλ_nis summable |A|.

Proof. As in the proof of Theorem 2, conditions (ix) and (x) of Theorem 2 imply conditions (i) and (ii) of Theorem 1.

A weighted mean matrix is a lower triangular matrix with entries a_nk = p_k/P_n, where {p_k} is a nonnegative sequence with p₀> 0 and P_n:= ∑ⁿ_k=0p_k. A weighted mean matrix is denoted by (N, p_n).

Corollary 1. Let {p_n} be a positive sequence such that P_n:= ∑ⁿ_k=0p_k→ ∞, and satisfies (xi) np_n= O(P_n).

If ∑anλ_nis bounded |N, p_n|_kand {λ_n} is a bounded nonzero sequence satisfying (xii)

∑

n=1

p_n

P_n|λ_n|^k= O(1), and (xiii) |∆λ_n| = O

µp_n P_n|λ_n|

¶ ,

then the series ∑anλ_nis summable |N, p_n|_k, k ≥ 1.

Proof. Conditions (i), (iv), and (v) of Theorem 1 are automatically satisfied for any weighted mean method.

Conditions (iii), (vii), and (viii) of Theorem 1 become, respectively, conditions (xi), (xii), and (xiii) of Corollary 1.

Corollary 2. If ∑anis bounded |N, p| and {λ_n} is a bounded nonzero sequence satisfying (a)

∑

p_n

P_n|λ_n| = O(1), and (b) P_n

p_n|∆λ_n| = O(|λ_n|),

then ∑a_nλ_nis summable |N, p|.

Proof. A weighted mean matrix automatically satisfies conditions (i)–(iii) of Theorem 1. Conditions (vii) and (viii) of Theorem 1 reduce to conditions (a) and (b) of Corollary 2, respectively.

Corollary 2 is a result of [3].

ACKNOWLEDGEMENTS

The second author acknowledges support from the Scientific and Technical Research Council of Turkey.

The authors thank the referees for their comments and suggestions.

REFERENCES

1. Rhoades, B. E. Inclusion theorems for absolute matrix summability methods. J. Math. Anal. Appl., 1999, 238, 82–90.

2. Rhoades, B. E. and Savas¸, E. Some necessary conditions for absolute matrix summability factors. Indian J. Pure Appl. Math., 2002, 33(7), 1003–1009.

3. Singh, N. On |N, pn| summability factors of infinite series. Indian J. Math., 1968, 10, 19–24.

L˜opmatute ridade |A|k-summeeruvusteguritest B. E. Rhoades ja Ekrem Savas¸

Olgu A kolmnurkne maatriks ja k ≥ 1. Artiklis on defineeritud rea |A|_k-summeeruvuse ja |A|_k-t˜okestatuse m˜oisted. On leitud piisavad tingimused selleks, et arvudλ_n oleksid maatriksi A k-j¨arku absoluutse sum- meeruvuse tegurid ehk rida ∑_na_nλ_n, kus (λ_n) on t˜okestatud jada, oleks |A|_k-summeeruv, kui rida ∑_na_n on

|A|_k-t˜okestatud. Saadud tulemus ¨uldistab N. Singhi tulemust Rieszi kaalutud keskmiste menetluse (N, p_n) absoluutse summeeruvuse tegurite kohta [3].