Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 675403, 10 pages doi:10.1155/2009/675403 Research Article A Recent Note on Quasi-Power Increasing Sequence for Generalized Absolute Summability E. Savas¸ 1 and H. S¸evli 2 1 Department of Mathematics, ˙ Istanbul Ticaret University, ¨ Usk ¨ udar, 34672- ˙ Istanbul, Turkey 2 Department of Mathematics, Faculty of Arts & Sciences, Y ¨ uz ¨ unc ¨ uYıl University, 65080-Van, Turkey Correspondence should be addressed to E. Savas¸, ekremsavas@yahoo.com Received 15 May 2009; Accepted 30 July 2009 Recommended by Ramm Mohapatra We prove two theorems on |A, δ| k , k ≥ 1, 0 ≤ δ<1/k, summability factors for an infinite series by using quasi-power increasing sequences. We obtain sufficient conditions for  a n λ n to be summable |A, δ| k , k ≥ 1, 0 ≤ δ<1/k, by using quasi-f -increasing sequences. Copyright q 2009 E. Savas¸andH.S¸evli. 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. 1. Introduction Quite recently, Savas¸ 1 obtained sufficient conditions for  a n λ n to be summable |A, δ| k , k ≥ 1, 0 ≤ δ<1/k. The purpose of this paper is to obtain the corresponding result for quasi- f-increasing sequence. Our result includes and moderates the conditions of his theorem with the special case μ  0. 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. The concept of absolute summability of order k ≥ 1 was defined by Flett 2 as follows. Let  a n denote a series with partial sums {s n }, and A a lower triangular matrix. Then  a n is said to be absolutely A-summable of order k ≥ 1, written that  a n is summable |A| k ,k ≥ 1, if ∞  n1 n k−1 | T n−1 − T n | k < ∞, 1.1 where T n  n  v0 a nv s v . 1.2 2 Journal of Inequalities and Applications In 3, Flett considered further extension of absolute summability in which he introduced a further parameter δ. The series  a n is said to be summable |A, δ| k , k ≥ 1, δ ≥ 0, if ∞  n1 n δkk−1 | T n−1 − T n | k < ∞. 1.3 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 4. 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 f or all n ≥ m ≥ 1. It should be noted that every almost increasing sequence is a 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 5.If1.4 stays with β  0, then γ is simply 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, 6. We may associate 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 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 . Quite recently, Savas¸ 1 obtained sufficient conditions for  a n λ n to be summable |A, δ| k , k ≥ 1, 0 ≤ δ<1/k as follows. Journal of Inequalities and Applications 3 Theorem 1.1. Let A be a lower triangular matrix with nonnegative entries satisfying a n−1,v ≥ a nv for n ≥ v  1, 1.7 a n0  1,n 0, 1, , 1.8 na nn  O  1  ,n−→ ∞ , 1.9 n−1  v1 a vv a n,v1  O  a nn  , 1.10 m1  nv1 n δk | Δ v a nv |  O  v δk a vv  , 1.11 m1  nv1 n δk a n,v1  O  v δk  , 1.12 and let {β n } and {λ n } be sequences such that | Δλ n | ≤ β n , 1.13 β n −→ 0,n−→ ∞ . 1.14 If {X n } is a quasi-β-power increasing sequence for some 0 <β<1 such that | λ n | X n  O  1  ,n−→ ∞ , 1.15 ∞  n1 nX n   Δβ n   < ∞, 1.16 m  n1 n δk−1 | s n | k  O  X m  ,m−→ ∞ , 1.17 then the series  a n λ n is summable |A, δ| k , k ≥ 1, 0 ≤ δ<1/k. Theorem 1.1 enhanced a theorem of Savas 7 by replacing an almost increasing sequence with a quasi-β-power increasing sequence for some 0 <β<1. It should be noted that if {X n } is an almost increasing sequence, then 1.15 implies that the sequence {λ n } is bounded. However, when {X n } is a quasi-β-power increasing sequence or a quasi-f- increasing sequence, 1.15 does not imply |λ m |  O1,m →∞. For example, since X m  m −β is a quasi-β-power increasing sequence for 0 <β<1 and if we take λ m  m δ , 0 <δ<β<1, then |λ m |X m  m δ−β  O1,m →∞holds but |λ m |  m δ /  O1see 8. Therefore, we remark that condition {λ n }∈bv 0 should be added to the statement of Theorem 1.1. The goal of this paper is to prove the following theorem by using quasi-f-increasing sequences. Our main result includes the moderated version of Theorem 1.1. We will show that the crucial condition of our proof, {λ n }∈bv 0 , can be deduced from another condition of the theorem. Also, we shall eliminate condition 1.15 in our theorem; however we shall deduce this condition from the conditions of our theorem. 4 Journal of Inequalities and Applications 2. The Main Results We now shall prove the following theorems. Theorem 2.1. Let A satisfy conditions 1.7–1.12, and let {β n } and {λ n } be sequences satisfying conditions 1.13 and 1.14 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 conditions 1.17 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, 0 ≤ δ<1/k, 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 . Theorem 2.1 includes the following theorem with the special case μ  0. Theorem 2.2 moderates the hypotheses of Theorem 1.1. Theorem 2.2. Let A satisfy conditions 1.7–1.12, and let {β n } and {λ n } be sequences satisfying conditions 1.13, 1.14, and 2.1.If{X n } is a quasi-β-power increasing sequence for some 0 ≤ β<1 and conditions 1.17 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, 0 ≤ δ<1/k. Remark 2.3. The crucial condition, {λ n }∈bv 0 , and condition 1.15 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 Theorem 2.1, also in the statement of Theorem 2.2 with the special case μ  0, conditions {λ n }∈bv 0 and 1.15 hold. 3. Lemmas We shall need the following lemmas for the proof of our main Theorem 2.1. Lemma 3.1 see 9. 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 Journal of Inequalities and Applications 5 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 8. 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 1.15 and λ n −→ 0,n−→ ∞ . 3.5 Lemma 3.3. 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 1.13, 1.14, 2.1, and 2.2, conditions 1.15 and 3.5 are satisfied. Proof. It is clear that 1.13 and 1.14⇒3.3.Also,1.13 and 2.2⇒3.4.ByLemma 3.2, under conditions 1.13-1.14 and 2.1–2.2, we have 1.15 and 3.5. Lemma 3.4. Let {X n } be a quasi-f-increasing sequence, where {f n }  {n β log n μ },μ ≥ 0, 0 ≤ β< 1. If conditions 1.13, 1.14, and 2.2 are satisfied, then nβ n X n  O  1  , 3.6 ∞  n1 β n X n < ∞. 3.7 Proof. It is clear that if {X n } is quasi-f-increasing, then {n β log n μ X n } is quasi-increasing. Since β n → 0,n →∞, from the fact that {n 1−β log n −μ } is increasing and 2.2, we have nβ n X n  nX n ∞  kn   Δβ k    O  1  n 1−β  log n  −μ ∞  kn k β  log k  μ X k   Δβ k    O  1  ∞  kn kX k   Δβ k    O  1  . 3.8 6 Journal of Inequalities and Applications Again using 2.2, ∞  n1 β n X n  O  1  ∞  n1 X n ∞  kn   Δβ k    O  1  ∞  k1   Δβ k   k  n1 n β  log n  μ X n n −β  log n  −μ  O  1  ∞  k1 k β  log k  μ X k   Δβ k   k  n1 n −β  log n  −μ  O  1  ∞  k1 kX k  β, μ    Δβ k    O  1  . 3.9 4. Proof of Theorem 2.1 Let y n denote the nth term of the A-transform of the series  a n λ n . Then, by definition, we have y n  n  i0 a ni s i  n  v0 a nv λ v a v . 4.1 Then, for n ≥ 1, we have Y n : y n − y n−1  n  v0 a nv λ v a v . 4.2 Applying Abel’s transformation, we may write Y n  n−1  v1 Δ v  a nv λ v  v  r1 a r  a nn λ n n  v1 a v . 4.3 Since Δ v  a nv λ v   λ v Δ v a nv Δλ v a n,v1 , 4.4 we have Y n  a nn λ n s n  n−1  v1 Δ v a nv λ v s v  n−1  v1 a n,v1 Δλ v s v  Y n,1  Y n,2  Y n,3 , say. 4.5 Journal of Inequalities and Applications 7 Since | Y n,1  Y n,2  Y n,3 | k ≤ 3 k  | Y n,1 | k  | Y n,2 | k  | Y n,3 | k  , 4.6 to complete the proof, it is sufficient to show that ∞  n1 n δkk−1 | Y n,r | k < ∞, for r  1, 2, 3. 4.7 Since {λ n } is bounded by Lemma 3.3,using1.9, we have I 1  m  n1 n δkk−1 | Y n,1 | k  m  n1 n δkk−1 | a nn λ n s n | k ≤ m  n1 n δk  na nn  k−1 a nn | λ n | k−1 | λ n || s n | k  O  1  m  n1 n δk a nn | λ n || s n | k . 4.8 Using properties 1.15,inviewofLemma 3.3,and3.7,from1.9, 1.13,and1.17, I 1  O  1  m−1  n1 | Δλ n | n  v1 v δk a vv | s v | k  O  1  | λ m | m  v1 v δk a vv | s v | k  O  1  m−1  n1 | Δλ n | n  v1 v δk−1 | s v | k  O  1  | λ m | m  v1 v δk−1 | s v | k  O  1  m−1  n1 β n X n  O  1  | λ m | X m  O  1  as m −→ ∞ . 4.9 Applying H ¨ older’s inequality, I 2  m1  n2 n δkk−1 | Y n,2 | k  O  1  m1  n2 n δkk−1  n−1  v1 | Δ v a nv || λ v || s v |  k  O  1  m1  n2 n δkk−1 n−1  v1 | Δ v a nv || λ v | k | s v | k  n−1  v1 | Δ v a nv |  k−1 . 4.10 8 Journal of Inequalities and Applications Using 1.9 and 1.11 and boundedness of {λ n }, I 2  O  1  m1  n2 n δk  na nn  k−1 n−1  v1 | Δ v a nv || s v | k | λ v | k−1 | λ v |  O  1  m  v1 | λ v || s v | k m1  nv1 n δk | Δ v a nv |  O  1  m  v1 v δk a vv | λ v || s v | k  O  1  , as m −→ ∞ , 4.11 as in the proof of I 1 . Finally, again using H ¨ older’s inequality, from 1.9, 1.10,and1.12, I 3  m1  n2 n δkk−1 | Y n,3 | k  O  1  m1  n2 n δkk−1  n−1  v1 a n,v1 | Δλ v || s v |  k  O  1  m1  n2 n δkk−1 n−1  v1 a n,v1 | Δλ v | k | s v | k a 1−k vv  n−1  v1 a vv a n,v1  k−1  O  1  m1  n2 n δk n−1  v1 a n,v1 | Δλ v | k | s v | k a 1−k vv  O  1  m  v1 | Δλ v | k | s v | k a 1−k vv m1  nv1 n δk a n,v1  O  1  m  v1  v | Δλ v |  k v δk a vv | s v | k . 4.12 By Lemma 3.1, condition 3.3,inviewofLemma 3.3, implies that n | Δλ n | ≤ 2n ∞  kn | Δ | Δλ k || ≤ 2 ∞  kn k | Δ | Δλ k || 4.13 holds. Thus, by Lemma 3.3, 3.4 implies that {n|Δλ n |} is bounded. Therefore, from 1.9 and 1.13, I 3  O  1  m  v1  v | Δλ v |  k−1 v | Δλ v | v δk a vv | s v | k  O  1  m  v1 vβ v v δk−1 | s v | k . 4.14 Journal of Inequalities and Applications 9 Using Abel transformation and 1.17, I 3  O  1  m−1  v1   Δ  vβ v     v  r1 r δk−1 | s r | k   O  1  mβ m m  v1 v δk−1 | s v | k  O  1  m−1  v1   Δ  vβ v    X v  O  1  mβ m X m . 4.15 Since Δ  vβ v   vβ v −  v  1  β v1  vΔβ v − β v1 , 4.16 we have I 3  O  1  m−1  v1 vX v   Δβ v    O  1  m−1  v1 X v1 β v1  O  1  mX m β m  O  1  , as m −→ ∞ , 4.17 by virtue of 2.2 and properties 3.6 and 3.7 of Lemma 3.4. So we obtain 4.7. This completes the proof. 5. Corollaries and Applications to Weighted Means Setting δ  0 in Theorems 2.1 and 2.2 yields the following two corollaries, respectively. Corollary 5.1. Let A satisfy conditions 1.7–1.10, and let {β n } and {λ n } be sequences satisfying conditions 1.13, 1.14, 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 | s 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 1.17 reduces condition 5.1 .Inthis case conditions 1.11 and 1.12 are obtained by conditions 1.7–1.10. Corollary 5.2. Let A satisfy conditions 1.7–1.10, and let {β n } and {λ n } be sequences satisfying conditions 1.13, 1.14, 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. A weighted mean matrix, denoted by  N,p n , is a lower triangular matrix with entries a nv  p v /P n , where {p n } is nonnegative sequence with p 0 > 0andP n :  n v0 p v →∞, as n →∞. 10 Journal of Inequalities and Applications Corollary 5.3. Let {p n } be a positive sequence satisfying 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 1.13, 1.14, 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 1.17 and 2.2 are satisfied, then the series,  a n λ n is summable |N,p n ,δ| k for k ≥ 1 and 0 ≤ δ<1/k. Proof. In Theorem 2.1 set A N,p n . It is clear that conditions 1.7, 1.8,and1.10 are automatically satisfied. Condition 1.9 becomes condition 5.2, and conditions 1.11 and 1.12 become condition 5.3 for weighted mean method. Corollary 5.3 includes the following result with the special case μ  0. Corollary 5.4. Let {p n } be a positive sequence satisfying 5.2 and 5.3, and let {X n } be a quasi-β- power increasing sequence for some 0 ≤ β<1. Then under conditions 1.13, 1.14, 1.17, 2.1, and 2.3,  a n λ n is summable |N,p n ,δ| k , k ≥ 1, 0 ≤ δ<1/k. References 1 E. Savas¸, “Quasi-power increasing sequence for generalized absolute summability,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 1, pp. 170–176, 2008. 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 T. M. Flett, “Some more theorems concerning the absolute summability of Fourier series and power series,” Proceedings of the London Mathematical Society, vol. 8, pp. 357–387, 1958. 4 S. Alijancic and D. Arendelovic, “O-regularly varying functions,” Publications de l’Institut Math ´ ematique, vol. 22, no. 36, pp. 5–22, 1977. 5 L. Leindler, “A new application of quasi power increasing sequences,” Publicationes Mathematicae Debrecen, vol. 58, no. 4, pp. 791–796, 2001. 6 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. 7 E. Savas¸, “On almost increasing sequences for generalized absolute summability,” Mathematical Inequalities & Applications, vol. 9, no. 4, pp. 717–723, 2006. 8 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. 9 L. Leindler, “A note on the absolute Riesz summability factors,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 4, article 96, 5 pages, 2005. . Mathematical Analysis and Applications, vol. 322, no. 2, pp. 1224–1230, 2006. 7 E. Savas¸, On almost increasing sequences for generalized absolute summability,” Mathematical Inequalities & Applications,. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 675403, 10 pages doi:10.1155/2009/675403 Research Article A Recent Note on Quasi-Power Increasing Sequence. 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, 6. We may associate A two lower triangular matrices A and  A as follows: a nv  n  rv a nr ,n,v