Volume 8 (2007), Issue 3, Article 82, 7 pp. A GENERAL NOTE ON INCREASING SEQUENCES HÜSEYİN BOR D EPARTMENT OF M ATHEMATICS E RCIYES U NIVERSITY 38039 K AYSERI , TURKEY bor@erciyes.edu.tr URL: http://fef.erciyes.edu.tr/math/hbor.htm Received 23 December, 2006; accepted 08 August, 2007 Communicated by S.S. Dragomir A BSTRACT. In the present paper, a general theorem on | N̄ , pn |k summability factors of infinite series has been proved under more weaker conditions. Also we have obtained a new result concerning the | C, 1 |k summability factors. Key words and phrases: Absolute summability, Summability factors, Almost and power increasing sequences, Infinite series. 2000 Mathematics Subject Classification. 40D15, 40F05, 40G99. 1. I NTRODUCTION A positive sequence (bn ) is said to be almost increasing if there exists a positive increasing sequence (cn ) and two positive constants A and B such that Acn ≤ bn ≤ Bcn (see [1]). We denote by BVO the expression BV ∩ CO , where CO and BV are the Pset of all null sequences and the set of all sequences with bounded variation, respectively. Let an be a given infinite series with partial sums (sn ). We denote by uαn and tαn the n-th Cesàro means of order α, with α > −1, of the sequences (sn ) and (nan ), respectively, i.e., n 1 X α−1 α A sv , (1.1) un = α An v=0 n−v tαn (1.2) n 1 X α−1 = α A vav , An v=1 n−v where Aαn = O(nα ), (1.3) The series (1.4) 009-07 P α > −1, Aα0 = 1 and Aα−n = 0 for n > 0. an is said to be summable |C, α|k , k ≥ 1, if (see [6, 8]) ∞ ∞ X k X |tαn |k nk−1 uαn − uαn−1 = < ∞. n n=1 n=1 2 H ÜSEYIN B OR If we take α = 1, then we get |C, 1|k summability. Let (pn ) be a sequence of positive numbers such that n X (1.5) Pn = pv → ∞ as n → ∞, (P−i = p−i = 0, i ≥ 1). v=0 The sequence-to-sequence transformation σn = (1.6) n 1 X p v sv Pn v=0 defines the sequence (σn ) of the Riesz mean or simply the (N̄ , pn )P mean of the sequence (sn ), generated by the sequence of coefficients (pn ) (see [7]). The series an is said to be summable N̄ , pn , k ≥ 1, if (see [2, 3]) k (1.7) ∞ X (Pn /pn )k−1 |∆σn−1 |k < ∞, n=1 where n pn X (1.8) ∆σn−1 = − Pv−1 av , n ≥ 1. Pn Pn−1 v=1 In the special case pn = 1 for all values of n, N̄ , pn k summability is the same as |C, 1|k summability. 2. K NOWN R ESULTS Mishra and Srivastava [10] have proved the following theorem concerning the N̄ , pn summability factors. Theorem A. Let (Xn ) be a positive non-decreasing sequence and let there be sequences (βn ) and (λn ) such that |∆λn | ≤ βn , (2.1) βn → 0 as (2.2) ∞ X (2.3) n → ∞, n |∆βn | Xn < ∞, n=1 |λn | Xn = O(1). (2.4) If (2.5) n X |sv | v=1 v = O(Xn ) as n→∞ and (pn ) is a sequence such that (2.6) Pn = O(npn ), (2.7) Pn ∆pn = O(pn pn+1 ), J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 82, 7 pp. http://jipam.vu.edu.au/ A G ENERAL N OTE ON I NCREASING S EQUENCES 3 P∞ n λn an Pnp is summable N̄ , pn . n Later on Bor [4] generalized Theorem A for N̄ , pn k summability in the following form. then the series n=1 Theorem B. Let (Xn ) be a positive non-decreasing sequence and the sequences (βn ) and (λn ) are such that conditions (2.1) – (2.7) of Theorem A are satisfied with the condition (2.5) replaced by: n X |sv |k (2.8) v=1 Then the series P∞ Pn λn n=1 an npn v = O(Xn ) as n → ∞. is summable N̄ , pn k , k ≥ 1. It may be noticed that if we take k = 1, then we get Theorem A. Quite recently Bor [5] has proved Theorem B under weaker conditions by taking an almost increasing sequence instead of a positive non-decreasing sequence. Theorem C. Let (Xn ) be an almost increasing sequence. If the conditions (2.1) – (2.4) and P Pn λn (2.6) – (2.8) are satisfied, then the series ∞ a is summable N̄ , p n k , k ≥ 1. n=1 n npn Remark 2.1. It should be noted that, under the conditions of Theorem B, (λn ) is bounded and ∆λn = O(1/n) (see [4]). 3. M AIN R ESULT The aim of this paper is to prove Theorem C under weaker conditions. For this we need the concept of quasi β-power increasing sequences. A positive sequence (γn ) is said to be a quasi β-power increasing sequence if there exists a constant K = K(β, γ) ≥ 1 such that Knβ γn ≥ mβ γm (3.1) holds for all n ≥ m ≥ 1. It should be noted that almost every 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 the example, say γn = n−β for β > 0. Now we shall prove the following theorem. Theorem 3.1. Let (Xn ) be a quasi β-power increasing sequence for some 0 < β < 1. If the conditions (2.1) – (2.4), (2.6) – (2.8) and (λn ) ∈ BVO (3.2) are satisfied, then the series P∞ n=1 n λn an Pnp is summable N̄ , pn k , k ≥ 1. n It should be noted that if we take (Xn ) as an almost increasing sequence, then we get Theorem C. In this case, condition (3.2) is not needed. We require the following lemma for the proof of Theorem 3.1. Lemma 3.2 ([9]). Except for the condition (3.2), under the conditions on (Xn ), (βn ) and (λn ) as taken in the statement of Theorem 3.1, the following conditions hold, when (2.3) is satisfied: (3.3) (3.4) nXn βn = O(1), ∞ X βn Xn < ∞. n=1 J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 82, 7 pp. http://jipam.vu.edu.au/ 4 H ÜSEYIN B OR Proof of Theorem 3.1. Let (Tn ) denote the (N̄ , pn ) mean of the series definition, we have Tn = (3.5) P∞ n=1 an Pn λn . npn Then, by n v n 1 X X ar P r λ r 1 X av P v λ v pv = (Pn − Pv−1 ) , Pn v=1 r=1 rpr Pn v=1 vpv and thus Tn − Tn−1 (3.6) n pn X Pv−1 Pv av λv , = Pn Pn−1 v=1 vpv n ≥ 1. Using Abel’s transformation, we get Tn − Tn−1 n Pv−1 Pv λv λn s n pn X = sv ∆ + Pn Pn−1 v=1 vpv n n−1 pn X Pv+1 Pv ∆λv sn λn = + sv n Pn Pn−1 v=1 (v + 1)pv+1 n−1 n−1 pn X Pv pn X 1 + P v s v λv ∆ − s v Pv λv Pn Pn−1 v 1 vpv Pn Pn−1 v=1 v = = Tn,1 + Tn,2 + Tn,3 + Tn,4 , say. To prove Theorem 3.1, by Minkowski’s inequality, it is sufficient to show that k−1 ∞ X Pn (3.7) pn n=1 |Tn,r |k < ∞, for r = 1, 2, 3, 4. Firstly by using Abel’s transformation, we have k−1 m X Pn n=1 pn k−1 m X Pn |sn |k |Tn,1 | = |λn |k−1 |λn | npn n n=1 k = O(1) m X n=1 = O(1) m−1 X n=1 = O(1) m−1 X |λn | | |sn | |k n ∆ |λn | n X |sv |k v=1 v + O(1) |λm | m X |sn |k n=1 n |∆λn | Xn + O(1) |λm | Xm n=1 = O(1) m−1 X βn Xn + O(1) |λm | Xm = O(1) as m → ∞, n=1 by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2. J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 82, 7 pp. http://jipam.vu.edu.au/ A G ENERAL N OTE ON I NCREASING S EQUENCES 5 Now, using the fact that Pv+1 = O((v + 1)pv+1 ), by (2.6), and then applying Hölder’s inequality, we have k m+1 n−1 m+1 X pn X X Pn k−1 k |Tn,2 | = O(1) Pv sv ∆λv k pn P P n=2 n n−1 v=1 n=2 ( )k m+1 n−1 X pn X Pv = O(1) |sv | pv |∆λv | p P Pk n=2 n n−1 v=1 v !k−1 k m+1 n−1 n−1 X pn X X Pv 1 = O(1) |sv |k pv |∆λv |k pv P P p P n n−1 v n−1 n=2 v=1 v=1 m m+1 k X Pv X pn = O(1) |sv |k pv |∆λv |k pv P P v=1 n=v+1 n n−1 m k−1 X Pv |∆λv | = O(1) |sv |k |∆λv | pv v=1 k−1 m X Pv k = O(1) |sv | |∆λv | vpv v=1 = O(1) m X v=1 = O(1) m−1 X vβv |sv |k v ∆(vβv ) v=1 = O(1) m−1 X v X |sr |k r=1 r + O(1)mβm m X |sv |k v=1 v |∆(vβv )| Xv + O(1)mβm Xm v=1 = O(1) m−1 X v |∆βv | Xv + O(1) v=1 m−1 X |βv | Xv + O(1)mβm Xm = O(1) v=1 as m → ∞, in view of the hypotheses of Theorem 3.1 and Lemma 3.2. Pv Again, since ∆( vp ) = O( v1 ), by (2.6) and (2.7) (see [10]), as in Tn,1 we have v ( n−1 )k m+1 m+1 X X Pn k−1 X 1 p n |Tn,3 |k = O(1) Pv |sv | |λv | k p v P P n n n−1 n=2 n=2 v=1 ( n−1 )k m+1 X pn X Pv 1 = O(1) pv |sv | |λv | pv v P Pk n=2 n n−1 v=1 ( )k−1 k m+1 n−1 n−1 X pn X Pv 1 X k k = O(1) pv |sv | |λv | pv P P vpv Pn−1 v=1 n=2 n n−1 v=1 k m m+1 X X Pv pn k k = O(1) |sv | pv |λv | vpv P P v=1 n=v+1 n n−1 k m X Pv 1 v = O(1) pv |sv |k |λv |k . vpv Pv v v=1 J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 82, 7 pp. http://jipam.vu.edu.au/ 6 H ÜSEYIN B OR = O(1) = O(1) k−1 m X Pv vpv v=1 m X |λv | v=1 = O(1) m−1 X k−1 |λv | |sv |k |λv | v |sv |k v Xv βv + O(1)Xm |λm | = O(1) as m → ∞. v=1 Finally, using Hölder’s inequality, as in Tn,3 we have k m+1 m+1 n−1 X Pn k−1 X pn X P v | Tn,4 |k = s λ v v k p v P P n n=2 n n−1 v=1 n=2 k m+1 n−1 X pn X Pv = s p λ v v v k vpv P P n=2 n n−1 v=1 m+1 X k n−1 Pv pn X k |sv | pv |λv |k ≤ P P vpv n=2 n n−1 v=1 m k X Pv 1 v = O(1) |sv |k pv |λv |k · vpv Pv v v=1 = O(1) m X |λv | v=1 = O(1) m−1 X 1 n−1 X Pn−1 v=1 !k−1 pv |sv |k v Xv βv + O(1)Xm |λm | = O(1) as m → ∞. v=1 Therefore we get k−1 m X Pn n=1 pn |Tn,r |k = O(1) as m → ∞, for r = 1, 2, 3, 4. This completes the proof of Theorem 3.1. Finally if we take pn = 1 for all values of n in the theorem, then we obtain a new result concerning the |C, 1|k summability factors. R EFERENCES [1] S. ALJANCIC (1977), 5–22. AND D. ARANDELOVIC, O-regularly varying functions, Publ. Inst. 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