Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 105136, 10 pages doi:10.1155/2010/105136 Research Article A Summability Factor Theorem for Quasi-Power-Increasing Sequences E. Savas¸ Department of Mathematics, ˙ Istanbul Ticaret University, ¨ Usk ¨ udar, 34378 ˙ Istanbul, Turkey Correspondence should be addressed to E. Savas¸, ekremsavas@yahoo.com Received 23 June 2010; Revised 3 September 2010; Accepted 15 September 2010 Academic Editor: J. Szabados Copyright q 2010 E. Savas¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We establish a summability factor theorem for summability |A, δ| k ,whereA is lower triangular matrix with nonnegative entries satisfying certain conditions. This paper is an extension of the main result of the work by Rhoades and Savas¸ 2006 by using quasi f-increasing sequences. 1. Introduction Recently, Rhoades and Savas¸ 1 obtained sufficient conditions for  a n λ n to be summable |A, δ| k , k ≥ 1 by using almost increasing sequence. The purpose of this paper is to obtain the corresponding result for quasi f-increasing sequence. A sequence {λ n } is said to be of bounded variation bv if  n |Δλ n | < ∞. Let bv 0  bv ∩ c 0 , where c 0 denotes the set of all null sequences. Let A be a lower triangular matrix, {s n } a sequence. Then A n : n  ν0 a nν s ν . 1.1 A series  a n , with partial sums s n , is said to be summable |A| k ,k ≥ 1if ∞  n1 n k−1 | A n − A n−1 | k < ∞, 1.2 and it is said to be summable |A, δ| k ,k ≥ 1andδ ≥ 0ifsee, 2 ∞  n1 n δkk−1 | A n − A n−1 | k < ∞. 1.3 2 Journal of Inequalities and Applications A positive sequence {b n } is said to be an almost increasing sequence if there exist an increasing sequence {c n } and positive constants A and B such that Ac n ≤ b n ≤ Bc n see, 3. Obviously, every increasing sequence is almost increasing. However, the converse need not be true as can be seen by taking the example, say b n  e −1 n n. A positive sequence γ : {γ n } is said to be a quasi β-power increasing sequence if there exists a constant K  Kβ, γ ≥ 1 such that Kn β γ n ≥ m β γ m 1.4 holds for all n ≥ m ≥ 1. It should be noted that every almost increasing sequence is quasi β-power increasing sequence for any nonnegative β, but the converse need not be true as can be seen by taking an example, say γ n  n −β for β>0 see, 4. A sequence satisfying 1.4 for β  0 is called a quasi-increasing sequence. It is clear that if {γ n } is quasi β-power increasing then {n β γ n } is quasi-increasing. A positive sequence γ  {γ n } is said to be a quasi-f-power increasing sequence if there exists a constant K  Kγ,f ≥ 1 such that Kf n γ n ≥ f m γ m holds for all n ≥ m ≥ 1, where f : {f n }  {n β log n μ },μ>0, 0 <β<1, see, 5. We may associate with A two lower triangular matrices A and  A as follows: a nv  n  rv a nr ,n,v 0, 1, , a nv  a nv − a n−1,v ,n 1, 2, , 1.5 where a 00  a 00  a 00 . 1.6 Given any sequence {x n }, the notation x n  O1 means that x n  O1 and 1/x n  O1. For any matrix entry a nv , Δ v a nv : a nv − a n,v1 . Rhoades and Savas¸ 1 proved the following theorem for |A, δ| k summability factors of infinite series. Theorem 1.1. Let {X n } be an almost increasing sequence and let {β n } and {λ n } be sequences such that i |Δλ n |≤β n , ii lim β n  0, iii  ∞ n1 n|Δβ n |X n < ∞, iv |λ n |X n  O1. Let A be a lower triangular matrix with nonnegative entries satisfying v na nn  O1, vi a n−1,ν ≥ a nν for n ≥ ν  1, vii a n0  1 for all n, Journal of Inequalities and Applications 3 viii  n−1 ν1 a νν a nν1  Oa nn , ix  m1 nν1 n δk |Δ ν a nν |  Oν δk a νν  and x  m1 nν1 n δk a nν1  Oν δk . If xi  m n1 n δk−1 |t n | k  OX m ,wheret n :1/n  1  n k1 ka k , then the series  a n λ n is summable |A, δ| k ,k ≥ 1. It should be noted that, if {X n } is an almost increasing sequence, then condition iv implies that the sequence {λ n } is bounded. However, if {X n } is a quasi β-power increasing sequence or a quasi f-increasing sequence, iv does not imply that λ is bounded. For example, the sequence {X m } defined by X m  m −β is trivially a quasi β-power increasing sequence for each β>0. If λ  {m δ }, for any 0 <δ<β,then λ m X m  m δ−β  O1, but λ is not bounded, see, 6, 7. The purpose of this paper is to prove a theorem by using quasi f-increasing sequences. We show that the crucial condition of our proof, {λ n }∈bv 0 , can be deduced from another condition of the theorem. 2. The Main Results We now will prove the following theorems. Theorem 2.1. Let A satisfy conditions (v)–(x) and let {β n } and {λ n } be sequences satisfying conditions (i) and (ii) of Theorem 1.1 and m  n1 λ n  o  m  ,m−→ ∞ . 2.1 If {X n } is a quasi f-increasing sequence and condition (xi) and ∞  n1 nX n  β, μ    Δβ n   < ∞ 2.2 are satisfied then the series  a n λ n is summable |A, δ| k , k ≥ 1, where {f n } : {n β log n μ },μ ≥ 0, 0 ≤ β<1, and X n β, μ :n β log n μ X n . The following theorem is the special case of Theorem 2.1 for μ  0. Theorem 2.2. Let A satisfy conditions (v)–(x) and let {β n } and {λ n } be sequences satisfying conditions (i), (ii), and 2.1.If{X n } is a quasi β-power increasing sequence for some 0 ≤ β<1 and conditions (xi) and ∞  n1 nX n  β    Δβ n   < ∞ 2.3 are satisfied, where X n β :n β X n , then the series  a n λ n is summable |A, δ| k , k ≥ 1. 4 Journal of Inequalities and Applications Remark 2.3. The conditions {λ n }∈bv 0 , and iv do not appear among the conditions of Theorems 2.1 and 2.2.ByLemma 3.3, under the conditions on {X n }, {β n },and{λ n } as taken in the statement of the Theorem 2.1, also in the statement of Theorem 2.2 with the special case μ  0, conditions {λ n }∈bv 0 and iv hold. 3. Lemmas We will need the following lemmas for the proof of our main Theorem 2.1. Lemma 3.1 see 8. Let {ϕ n } be a sequence of real numbers and denote Φ n : n  k1 ϕ k , Ψ n : ∞  kn   Δϕ k   . 3.1 If Φ n  on then there exists a natural number N such that   ϕ n   ≤ 2Ψ n 3.2 for all n ≥ N. Lemma 3.2 see 9. If {X n } is a quasi f-increasing sequence, where {f n }  {n β log n μ },μ ≥ 0, 0 ≤ β<1, then conditions 2.1 of Theorem 2.1, m  n1 | Δλ n |  o  m  ,m−→ ∞ , 3.3 ∞  n1 nX n  β, μ  | Δ | Δλ n || < ∞, 3.4 where X n β, μn β log n μ X n , imply conditions (iv) and λ n −→ 0,n−→ ∞ . 3.5 Lemma 3.3 see 7. If {X n } is a quasi f-increasing sequence, where {f n }  {n β log n μ },μ ≥ 0, 0 ≤ β<1, then under conditions (i), (ii), 2.1 , and 2.2, conditions (iv) and 3.5 are satisfied. Lemma 3.4 see 7. Let {X n } be a quasi f-increasing sequence, where {f n }  {n β log n μ }, μ ≥ 0, 0 ≤ β<1. If conditions (i), (ii), and 2.2 are satisfied, then nβ n X n  O  1  , 3.6 ∞  n1 β n X n < ∞. 3.7 Journal of Inequalities and Applications 5 4. Proof of Theorem 2.1 Proof. Let y n  be the nth term of the A transform of the partial sums of  n i0 λ i a i . Then we have y n : n  i0 a ni s i  n  i0 a ni i  ν0 λ ν a ν  n  ν0 λ ν a ν n  iν a ni  n  ν0 a nν λ ν a ν , 4.1 and, for n ≥ 1, we have Y n : y n − y n−1  n  ν0 a nν λ ν a ν . 4.2 We may write noting that vii implies that a n0  0, Y n  n  ν1  a nν λ ν ν  νa ν  n  ν1  a nν λ ν ν   ν  r1 ra r − ν−1  r1 ra r   n−1  ν1 Δ ν  a nν λ ν ν  ν  r1 ra r  a nn λ n n n  r1 ra r  n−1  ν1  Δ ν a nν  λ ν ν  1 ν t ν  n−1  ν1 a n,ν1  Δλ ν  ν  1 ν t ν  n−1  ν1 a n,ν1 λ ν1 1 ν t ν   n  1  a nn λ n t n n  T n1  T n2  T n3  T n4 , say. 4.3 To complete the proof it is sufficient, by Minkowski’s inequality, to show that ∞  n1 n δkk−1 | T nr | k < ∞, for r  1, 2, 3, 4. 4.4 From the definition of  A and using vi and vii it follows that a n,ν1 ≥ 0. 4.5 6 Journal of Inequalities and Applications Using H ¨ older’s inequality I 1 : m  n1 n δkk−1 | T n1 | k  m  n1 n δkk−1      n−1  ν1 Δ ν a nν λ ν ν  1 ν t ν      k  O  1  m1  n1 n δkk−1  n−1  ν1 | Δ ν a nν || λ ν || t ν |  k  O  1  m1  n1 n δkk−1  n−1  ν1 | Δ ν a nν || λ ν | k | t ν | k  n−1  ν1 | Δ ν a nν |  k−1 , Δ ν a nν  a nν − a 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. 4.6 Thus, using vii, n−1  ν0 | Δ ν a nν |  n−1  ν0  a n−1,ν − a nν   1 − 1  a nn  a nn . 4.7 Since λ n  is bounded by Lemma 3.3,usingv, ix, xi, i, and condition 3.7 of Lemma 3.4 I 1  O  1  m1  n1 n δk  na nn  k−1 n−1  ν1 | λ ν | k | t ν | k | Δ ν a nν |  O  1  m1  n1 n δk  n−1  ν1 | λ ν | k−1 | λ ν || Δ ν a nν || t ν | k   O  1  m  ν1 | λ ν || t ν | k m1  nν1 n δk | Δ ν a nν |  O  1  m  ν1 ν δk | λ ν | a νν | t ν | k  O  1  m  ν1 ν δk−1 | λ ν || t ν | k  O  1   m−1  ν1 Δ  | λ ν |  ν  r1 r δk−1 | t r | k  | λ m | m  r1 r δk−1 | t r | k   O  1  m−1  ν1 | Δλ ν | X ν  O  1  | λ m | X m  O  1  m  ν1 β ν X ν  O  1  | λ m | X m  O  1  . 4.8 Journal of Inequalities and Applications 7 Using H ¨ older’s inequality, I 2 : m1  n2 n δkk−1 | T n2 | k  m1  n2 n δkk−1      n−1  ν1 a n,ν1  Δλ ν  ν  1 ν t ν      k  O  1  m1  n2 n δkk−1  n−1  ν1 a n,ν1 | Δλ ν || t ν |  k  O  1  m1  n2 n δkk−1  n−1  ν1 | Δλ ν || t ν | k a n,ν1  n−1  ν1 a n,ν1 | Δλ ν |  k−1 . 4.9 By Lemma 3.1, condition 3.3,inviewofLemma 3.3 implies that ∞  n1 | Δλ n | ≤ 2 ∞  n1 ∞  kn | Δ | Δλ k ||  2 ∞  k1 | Δ | Δλ k || 4.10 holds. Thus by Lemma 3.3, 3.4 implies that  ∞ n1 |Δλ n | converges. Therefore, there exists a positive constant M such that  ∞ n1 |Δλ n |≤M and from the properties of matrix A,weobtain n−1  ν1 a n,ν1 | Δλ k | ≤ Ma nn . 4.11 We have, using v and x, I 2  O  1  m1  n2 n δk  na nn  k−1 n−1  ν1 a n,ν1 β ν | t ν | k  O  1  m  ν1 β ν | t ν | k m1  nν1 n δk a n,ν1 . 4.12 Therefore, I 2  O  1  m  ν1 ν δk β ν | t ν | k  O  1  m  ν1 νβ ν | t ν | k ν ν δk . 4.13 8 Journal of Inequalities and Applications Using summation by parts, 2.2, xi, and condition 3.6 and 3.7 of Lemma 3.4 I 2 : O  1  m−1  ν1 Δ  νβ ν  ν  r1 r δk−1 | t r | k  O  1  mβ m m  r1 r δk−1 | t r | k  O  1  m−1  ν1 ν   Δ  β ν    X ν  O  1  m−1  ν1 β ν1 X ν1  O  1   O  1  . 4.14 Using H ¨ older’s inequality and viii, m1  n2 n k−1 | T n3 | k  m1  n2 n δkk−1      n−1  ν1 a n,ν1 λ ν1 1 ν t ν      k ≤ m1  n2 n δkk−1  n−1  ν1 | λ ν1 | a n,ν1 ν | t ν |  k  O  1  m1  n2 n δkk−1  n−1  ν1 | λ ν1 | a n,ν1 | t ν | a νν  k  O  1  m1  n2 n δkk−1  n−1  ν1 | λ ν1 | k a νν | t ν | k a n,ν1  n−1  ν1 a νν | a n,ν1 |  k−1 . 4.15 Using boundedness of {λ n }, v, x, xi, Lemmas 3.3 and 3.4 I 3  O  1  m1  n2 n δk  na nn  k−1 n−1  ν1 | λ ν1 | k a νν | t ν | k a n,ν1  O  1  m  ν1 | λ ν1 | a νν | t ν | k m1  nν1 n δk a n,ν1  O  1  m  ν1 | λ ν1 | ν δk a νν | t ν | k  O  1  m  v1 | λ v1 |  va vv  v δk−1 | t v | k  O  1  m  v1 | λ v1 | v δk−1 | t v | k . 4.16 Journal of Inequalities and Applications 9 Using summation by parts I 3  O  1  m−1  v1 | Δλ v1 | v  r1 r δk−1 | t r | k  O  1  | λ m1 | m  v1 v δk−1 | t v | k  O  1  m−1  v1 | Δλ v1 | v1  r1 r δk−1 | t r | k  O  1  | λ m1 | m1  v1 v δk−1 | t v | k  O  1  m−1  v1 | Δλ v1 | X v1  O  1  | λ m1 | X m1  O  1  m−1  v1 β v1 X v1  O  1  | λ m1 | X m1  O  1  . 4.17 Finally, using boundedness of {λ n },andv we have m  n1 n δkk−1 | T n4 | k  m  n1 n δkk−1      n  1  a nn λ n t n n     k  O  1  m  n1 n δk a nn | λ n || t n | k  O  1  , 4.18 as in the proof of I 1 . 5. Corollaries and Applications to Weighted Means Setting δ  0inTheorem 2.1 and Theorem 2.2 yields the following two corollaries, respectively. Corollary 5.1. Let A satisfy conditions (v)–(viii) and let {β n } and {λ n } be sequences satisfying conditions (i), (ii), and 2.1.If{X n } is a quasi f-increasing sequence, where {f n } : {n β log n μ },μ≥ 0, 0 ≤ β<1, and conditions 2.2 and m  n1 1 n | t n | k  O  X m  ,m−→ ∞ , 5.1 are satisfied then the series  a n λ n is summable |A| k ,k ≥ 1. Proof. If we take δ  0inTheorem 2.1 then condition xi reduces condition 5.1. Corollary 5.2. Let A satisfy conditions (v)–(viii) and let {β n } and {λ n } be sequences satisfying conditions (i), (ii), and 2.1.If{X n } is a quasi β-power increasing sequence for some 0 ≤ β<1 and conditions 2.3 and 5.1 are satisfied then the series  a n λ n is summable |A| k , k ≥ 1. 10 Journal of Inequalities and Applications Corollary 5.3. Let {p n } be a positive sequence such that P n :  n i0 p i →∞, as n →∞satisfies np n  O  P n  , as n −→ ∞ , 5.2 m1  nv1 n δk p n P n P n−1  O  v δk P v  5.3 and let {β n } and {λ n } be sequences satisfying conditions (i), (ii), and 2.1.If{X n } is a quasi f- increasing sequence, where {f n } : {n β log n μ }, μ ≥ 0, 0 ≤ β<1, and conditions (xi) and 2.2 are satisfied then the series,  a n λ n is summable |N, p n ,δ| k for k ≥ 1. Proof. In Theorem 2.1,setA N,p n . Conditions i and ii of Corollary 5.3 are, respectively, conditions i and ii of Theorem 2.1. Condition v becomes condition 5.2 and conditions ix and x become condition 5.3 for weighted mean method. Conditions vi, vii,andviii of Theorem 2.1 are automatically satisfied for any weighted mean method. The following Corollary is the special case of Corollary 5.3 for μ  0. Corollary 5.4. Let {p n } be a positive sequence satisfying 5.2, 5.3 and let {X n } be a quasi β-power increasing sequence for some 0 ≤ β<1. Then under conditions (i), (ii), (xi), 2.1, and 2.3,  a n λ n is summable |N, p n ,δ| k ,k ≥ 1. References 1 B. E. Rhoades and E. Savas¸, “A summability factor theorem for generalized absolute summability,” Real Analysis Exchange, vol. 31, no. 2, pp. 355–363, 2006. 2 T. M. Flett, “On an extension of absolute summability and some theorems of Littlewood and Paley,” Proceedings of the London Mathematical Society , vol. 7, pp. 113–141, 1957. 3 S. Alijancic and D. Arendelovic, “O-regularly varying functions,” Publications de l’Institut Math ´ ematique , vol. 22, no. 36, pp. 5–22, 1977. 4 L. Leindler, “A new application of quasi power increasing sequences,” Publicationes Mathematicae Debrecen, vol. 58, no. 4, pp. 791–796, 2001. 5 W. T. Sulaiman, “Extension on absolute summability factors of infinite series,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 1224–1230, 2006. 6 E. Savas¸, “A note on generalized |A| k -summability factors for infinite series,” Journal of Inequalities and Applications, vol. 2010, Article ID 814974, 10 pages, 2010. 7 E. Savas¸andH.S¸evli, “A recent note on quasi-power increasing sequence for generalized absolute summability,” Journal of Inequalities and Applications, vol. 2009, Article ID 675403, 10 pages, 2009. 8 L. Leindler, “A note on the absolute Riesz summability factors,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 4, article 96, 2005. 9 H. S¸evli and L. Leindler, “On the absolute summability factors of infinite series involving quasi-power- increasing sequences,” Computers & Mathematics with Applications, vol. 57, no. 5, pp. 702–709, 2009. . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 105136, 10 pages doi:10.1155/2010/105136 Research Article A Summability Factor Theorem for Quasi-Power-Increasing. 2.1, and 2.3,  a n λ n is summable |N, p n ,δ| k ,k ≥ 1. References 1 B. E. Rhoades and E. Savas¸, A summability factor theorem for generalized absolute summability, ” Real Analysis Exchange,. 2006. 6 E. Savas¸, A note on generalized |A| k -summability factors for infinite series,” Journal of Inequalities and Applications, vol. 2010, Article ID 814974, 10 pages, 2010. 7 E. Savas¸andH.S¸evli,