Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 86095, 8 pages doi:10.1155/2007/86095 Research Article ANoteon |A| k Summability Factors for Infinite Series Ekrem Savas¸ and B. E. Rhoades Received 9 November 2006; Accepted 29 March 2007 Recommended by Martin J. Bohner We obtain sufficient conditions on a nonnegative lower triangular matrix A and a se- quence λ n for the series a n λ n /na nn to be absolutely summable of order k ≥ 1byA. Copyright © 2007 E. Savas¸ and B. E. Rhoades. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited. A weighted mean matrix, denoted by ( N, p n ), is a lower triangular matrix with entries p k /P n ,where{p k } is a nonnegative sequence with p 0 > 0, and P n := n k =0 p k . Mishra and Srivastava [1]obtainedsufficient conditions on a sequence {p k } and a sequence {λ n } for the series a n P n λ n /np n to be absolutely summable by the weighted mean matrix ( N, p n ). Bor [2] extended this result to absolute summability of order k ≥ 1. Unfortunately, an incorrect definition of absolute summability was used. In this note, we establish the corresponding result for a nonnegative triangle, using the correct definition of absolute summability of order k ≥ 1, (see [3]). As a corollary, we obtain the corrected version of Bor’s result. Let A be an infinite lower triangular matr ix. We may associate with A two lower trian- gular matrices A and A, whose entries are defined by a nk = n i=k a ni , a nk = a nk − a n−1,k ,(1) respectively. The motivation for these definitions will become clear as we proceed. Let A be an infinite matrix. The series a k is said to be absolutely summable by A,of order k ≥ 1, written as |A| k ,if ∞ k=0 n k−1 Δt n−1 k < ∞,(2) 2 Journal of Inequalities and Applications where Δ is the forward differ ence operator and t n denotes the nth term of the matrix transform of the sequence {s n },wheres n := n k =0 a k . Thus t n = n k=0 a nk s k = n k=0 a nk k ν=0 a ν = n ν=0 a ν n k=ν a nk = n ν=0 a nν a ν , t n − t n−1 = n ν=0 a nν a ν − n−1 ν=0 a n−1,ν a ν = n ν=0 a nν a ν , (3) since a n−1,n = 0. Theresulttobeprovedisthefollowing. Theorem 1. Let A be a triangle with nonnegative entries satisfy ing (i) a n0 = 1, n = 0,1, , (ii) a n−1,ν ≥ a nν for n ≥ ν +1, (iii) na nn O(1), (iv) Δ(1/a nn ) = O(1), (v) n ν =0 a νν |a n,ν+1 |=O(a nn ). If {X n } is a positive nondecreasing sequence and the sequences {λ n } and {β n } satisfy (vi) |Δλ n |≤β n , (vii) limβ n = 0, (viii) |λ n |X n = O(1), (ix) ∞ n=1 nX n |Δβ n | < ∞, (x) T n := n ν =1 (|s ν | k /ν) = O(X n ), then the series ∞ ν=1 a n λ n /na nn is summable |A| k , k ≥ 1. The proof of the theorem requires the following lemma. Lemma 2 (see Mishra and Srivastava [1]). Let {X n } be a positive nondecreasing sequence and the sequences {β n }, {λ n } satisfy conditions (vi)–(ix) of Theorem 1. Then nX n β n = O(1), (4) ∞ n=1 β n X n < ∞. (5) Since {X n } is nondecreasing, X n ≥ X 0 , which is a positive constant. Hence condition (viii) implies that λ n is bounded. It also follows from (4)thatβ n = O(1/n), and hence that Δλ n = O(1/n) by condition (iv). Proof. Let T n denote the nth term of the A-transform of the series (a n λ n )/(na nn ). Then we may write T n = n ν=0 a nν ν i=0 a i λ i a ii i = m i=0 a i λ i a ii i n ν=i a nν = n i=0 a ni a i λ i a ii i . (6) E. Savas¸ and B. E. Rhoades 3 Thus, T n − T n−1 = n i=0 a ni a i λ i a ii i − n−1 i=0 a n−1,i a i λ i a ii i = n i=0 a ni − a n−1,i a i λ i a ii i = n i=0 a ni a i λ i a ii i = n i=0 a ni λ i a ii i s i − s i−1 = n−1 i=0 a ni λ i a ii i s i + a nn λ n a nn n s n − n i=0 a ni λ i s i−1 a ii i = n−1 i=0 a ni λ i a ii i s i + a nn λ n a nn n s n − n−1 i=0 a n,i+1 λ i+1 s i (i +1)a i+1,i+1 = n i=0 a ni λ i a ii i − a n,i+1 λ i+1 (i +1)a i+1,i+1 s i + a nn λ n na nn . (7) We may wr i te a ni λ i ia ii − a n,i+1 λ i+1 (i +1)a i+1,i+1 = a ni λ i ia ii − a n,i+1 λ i+1 (i +1)a i+1,i+1 + a n,i+1 λ i (i +1)a i+1,i+1 − a n,i+1 λ i (i +1)a i+1,i+1 = Δ i a ni ia ii λ i + a n,i+1 (i +1)a i+1,i+1 Δ λ i . (8) Also we may write Δ i a ni ia ii λ i = a ni ia ii λ i − a n,i+1 (i +1)a i+1,i+1 λ i − a n,i+1 ia ii λ i + a n,i+1 ia ii λ i = Δ i a ni λ i ia ii + a n,i+1 λ i 1 ia ii − 1 (i +1)a i+1,i+1 . (9) Hence, T n − T n−1 = n−1 i=0 Δ i a ni ia ii λ i s i + n−1 i=0 a n,i+1 λ i 1 ia ii − 1 (i +1)a i+1,i+1 s i + n−1 i=0 a n,i+1 (i +1)a i+1,i+1 Δ i (λ i )s i + λ n n s n = T n1 + T n2 + T n3 + T n4 ,say. (10) To finish the proof of the theorem, it will be sufficient to show that ∞ n=1 n k−1 T nr k < ∞,forr = 1,2,3,4. (11) 4 Journal of Inequalities and Applications Using H ¨ older’s inequality and (iii), I 1 = m+1 n=1 n k−1 T n1 k ≤ m+1 n=1 n k−1 n−1 i=0 Δ i a ni ia ii λ i s i k = O(1) m+1 n=1 n k−1 n−1 i=0 Δ i a ni λ i s i k = O(1) m+1 n=1 n k−1 n−1 i=0 Δ i a ni λ i k s i k n−1 i=0 Δ i a ni k−1 . (12) But using (ii), Δ i a ni = a ni − a n,i+1 = a ni − a n−1,i − a n,i+1 + a n−1,i+1 = a ni − a n−1,i ≤ 0. (13) Thus using (i), n−1 i=0 Δ i a ni = n−1 i=0 a n−1,i − a ni = 1 − 1+a nn = a nn . (14) From (viii), it follows that λ n = O(1). Using (iii), (vi), (x), and property (5)of Lemma 2, I 1 = O(1) m+1 n=1 na nn k−1 n −1 i=0 λ i k s i k Δ i a ni = O(1) m+1 n=1 na nn k−1 n−1 i=0 λ i k−1 λ i Δ i a ni s i k = O(1) m i=0 λ i s i k m+1 n=i+1 na nn k−1 Δ i a ni = O(1) m i=0 λ i s i k a ii = λ 0 s 0 k a 00 + O(1) m i=1 λ i s i k i = O(1) + O(1) m i=1 λ i i r=1 s r k r − i−1 r=1 s r k r = O(1) m i=1 λ i i r=1 s r k r − m−1 j=0 λ j+1 j r=1 s r k r = O(1) m−1 i=1 Δ λ i i r=1 1 r s r k + O(1) λ m m i=1 s i k i E. Savas¸ and B. E. Rhoades 5 = O(1) m−1 i=1 Δ λ i X i + O(1) λ m X m = O(1) m i=1 β i X i + O(1) λ m X m = O(1), I 2 = m+1 n=1 n k−1 T n2 k = m+1 n=1 n k−1 n−1 i=0 a n,i+1 λ i Δ 1 ia ii s i k = O(1) m+1 n=1 n k−1 n−1 i=0 a n,i+1 λ i Δ 1 ia ii s i k . (15) Now Δ 1 ia ii = 1 ia ii − 1 (i +1)a i+1,i+1 = 1 ia ii − 1 (i +1)a i+1,i+1 + 1 (i +1)a ii − 1 (i +1)a ii = 1 (i +1) 1 a ii − 1 a i+1,i+1 + 1 a ii 1 i − 1 i +1 = 1 (i +1) Δ 1 a ii + 1 ia ii . (16) Thus using (iv) and (ii), Δ 1 ia ii = 1 i +1 Δ 1 a ii + 1 ia ii ≤ 1 i +1 a i+1,i+1 − a ii a ii a i+1,i+1 + 1 ia ii = 1 i +1 O(1) + O(1) . (17) Hence, using H ¨ older’s inequality, (v) and (iii), I 2 = O(1) m+1 n=1 n k−1 n−1 i=0 a n,i+1 λ i 1 i +1 s i k = O(1) m+1 n=1 n k−1 n−1 i=0 a n,i+1 a ii λ i s i k = O(1) m+1 n=1 n k−1 n−1 i=0 a n,i+1 a ii λ i k s i k n−1 i=0 a ii a n,i+1 k−1 = O(1) m+1 n=1 na nn k−1 n −1 i=0 a n,i+1 a ii λ i k s i k 6 Journal of Inequalities and Applications = O(1) m i=0 λ i k s i k a ii m+1 n=i+1 na nn k−1 a n,i+1 = O(1) m i=0 λ i k s i k a ii m+1 n=i+1 a n,i+1 . (18) From [4], m+1 n=i+1 a n,i+1 ≤ 1. (19) Hence, I 2 = O(1) m i=1 λ i k s i k a ii = O(1) m i=1 λ i λ i k−1 s i k 1 i = m i=1 λ i s i k i = O(1), (20) as in the proof of I 1 . Using (iii), H ¨ older’s inequalit y, and (v), I 3 = m+1 n=1 n k−1 T n3 k = m+1 n=1 n k−1 n−1 i=0 a n,i+1 Δλ i s i (i +1)a i+1,i+1 k = O(1) m+1 n=1 n k−1 n−1 i=0 a n,i+1 Δλ i s i k = O(1) m+1 n=1 n k−1 n−1 i=0 a ii a ii a n,i+1 Δλ i s i k = O(1) m+1 n=1 n k−1 n−1 i=0 a ii a n,i+1 a k ii Δλ i k s i k n−1 i=0 a ii a n,i+1 k−1 = O(1) m+1 n=1 na nn k−1 n −1 i=0 a ii a n,i+1 a k ii Δλ i k s i k = O(1) m+1 n=1 n −1 i=0 a n,i+1 Δλ i k s i k 1 a k ii a ii = O(1) m i=0 a ii a k ii Δλ i k s i k m+1 n=i+1 a n,i+1 = O(1) m i=0 Δλ i a ii k−1 Δλ i s i k = O(1) m i=0 Δλ i s i k = O(1) m i=0 s i k β i . (21) E. Savas¸ and B. E. Rhoades 7 Since |s i | k = i(T i − T i−1 )by(x),wehave I 3 = O(1) m i=1 i T i − T i−1 β i . (22) Using Abel’s transformation, (vi), and (5), I 3 = O(1) m−1 i=1 T i Δ iβ i + O(1)mT n β n = O(1) m−1 i=1 i Δβ i X i + O(1) m−1 i=1 X i β i + O(1)mX n β n = O(1). (23) Using (viii) and (x), I 4 = m+1 n=1 n k−1 T n4 k = m+1 n=1 n k−1 s n λ n n k = m+1 n=1 s n k λ n k 1 n = m+1 n=1 s n k n λ n λ n k−1 = O(1), (24) as in the proof of I 1 . Corollary 3. Let {p n } be a positive sequence such that P n = n k =0 p k →∞and satisfies (i) np n O(P n ); (ii) Δ(P n /p n ) = O(1). If {X n } is a positive nondecreasing sequence and the sequences {λ n } and {β n } are such that (iii) |Δλ n |≤β n , (iv) β n → 0 as n →∞, (v) |λ n |X n = O(1) as n →∞, (vi) ∞ n=1 nX n |Δβ n | < ∞, (vii) T n = n ν =1 |s ν | k /ν = O(X n ), then the series (a n P n λ n )/(np n ) is summable |N, p n | k , k ≥ 1. Proof. Conditions (iii)–(vii) of Corollary 3 are, respectively, conditions (vi)–(x) of Theo- rem 1 . Conditions (i), (ii), and (v) of Theorem 1 are automatically satisfied for any weighted mean method. Conditions (iii) and (iv) of Theorem 1 become, respectively, conditions (i) and (ii) of Corollary 3. Acknowledgment The first author received support from the Scientific and Technical Research Council of Turkey. 8 Journal of Inequalities and Applications References [1] K. N. Mishra and R. S. L. Srivastava, “On | –– N , p n | summability factors of infinite series,” Indian Journal of Pure and Applied Mathematics, vol. 15, no. 6, pp. 651–656, 1984. [2] H.Bor,“Anoteon | –– N , p n | k summability factors of infinite series,” Indian Journal of Pure and Applied Mathematics, vol. 18, no. 4, pp. 330–336, 1987. [3] T. M. Flett, “On an extension of absolute summability and some theorems of Littlewood and Paley,” Proceedings of the London Mathematical Society. Third Series, vol. 7, pp. 113–141, 1957. [4] B. E. Rhoades and E. Savas¸, “A note on absolute summability factors,” Periodica Mathematica Hungarica, vol. 51, no. 1, pp. 53–60, 2005. Ekrem Savas¸: Department of Mathematics, Faculty of Sciences and Arts, Istanbul Ticaret University, Uskudar, 34672 Istanbul, Turkey Email addresses: ekremsavas@yahoo.com; esavas@iticu.edu.tr B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA Email address: rhoades@indiana.edu . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 86095, 8 pages doi:10.1155/2007/86095 Research Article ANoteon |A| k Summability Factors for Infinite. Rhoades and E. Savas¸, A note on absolute summability factors, ” Periodica Mathematica Hungarica, vol. 51, no. 1, pp. 53–60, 2005. Ekrem Savas¸: Department of Mathematics, Faculty of Sciences and. Scientific and Technical Research Council of Turkey. 8 Journal of Inequalities and Applications References [1] K. N. Mishra and R. S. L. Srivastava, On | –– N , p n | summability factors of infinite