IJMMS 25:5 (2001) 293–298 PII. S0161171201005105 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON ALMOST INCREASING SEQUENCES AND ITS APPLICATIONS HIKMET S. ÖZARSLAN (Received 7 July 1999 and in revised form 18 April 2000) Abstract. We prove a general theorem on |N̄, pn ; δ|k summability factors, which generalizes a theorem of Bor (1994) on |N̄, pn |k summability factors, under weaker conditions by using an almost increasing sequence. 2000 Mathematics Subject Classification. Primary 40D15, 40F05, 40G99. 1. Introduction. Let an be a given infinite series with partial sums (sn ). Let (pn ) be a sequence of positive numbers such that Pn = n pv → ∞ as n → ∞, P−i = p−i = 0, i ≥ 1 . (1.1) v=0 The sequence-to-sequence transformation Tn = n 1 pv s v Pn v=0 (1.2) defines the sequence (Tn ) of the (N̄, pn ) mean of the sequence (sn ), generated by the sequence of coefficients (pn ) (see [4]). The series an is said to be summable |N̄, pn |k , k ≥ 1, if (see [1]) ∞ Pn k−1 |∆Tn−1 |k < ∞ (1.3) pn n=1 and it is said to be summable |N̄, pn ; δ|k , k ≥ 1 and δ ≥ 0, if (see [2]) ∞ Pn δk+k−1 |∆Tn−1 |k < ∞, pn n=1 (1.4) where ∆Tn−1 = − n pn Pv−1 av , Pn Pn−1 v=1 n ≥ 1. (1.5) In the special case when δ = 0, (respectively, pn = 1 for all values of n) |N̄, pn ; δ|k summability is the same as |N̄, pn |k (respectively, |C, 1; δ|k ) summability. 294 HIKMET S. ÖZARSLAN Mishra and Srivastava [6] proved the following theorem for |C, 1|k summability. Theorem 1.1. Let (Xn ) be a positive nondecreasing sequence and let there be sequences (βn ) and (λn ) such that ∆λn ≤ βn , βn → 0 as n → ∞, λn Xn = O(1) as n → ∞, ∞ n∆βn Xn < ∞. (1.6) (1.7) (1.8) (1.9) n=1 If m 1 sn k = O Xm n n=1 then the series as m → ∞, (1.10) an λn is summable |C, 1|k , k ≥ 1. Bor [3] has generalized Theorem 1.1 for |N̄, pn |k summability in the form of the following theorem. Theorem 1.2. Let (Xn ) be a positive nondecreasing sequence and the sequences (βn ) and (λn ) such that conditions (1.6), (1.7), (1.8), and (1.9) of Theorem 1.1 are satisfied. Furthermore, if (pn ) is a sequence of positive numbers such that Pn = O npn as n → ∞, m pn |sn |k = O Xm P n n=1 then the series as m → ∞, (1.11) (1.12) an λn is summable |N̄, pn |k , k ≥ 1. It should be noted that, if we take pn = 1 for all values of n, then condition (1.12) will be reduced to condition (1.10). Also, it can be noticed that in this case condition (1.11) is obvious. 2. The main result. The aim of this paper is to generalize Theorem 1.2 for |N̄, pn ; δ|k summability under weaker conditions. For this we need the concept of almost increasing sequence. 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 . Obviously every increasing sequence is almost increasing sequence n but the converse need not be true as can be seen from the example bn = ne(−1) . So we are weakening the hypotheses of Theorem 1.2 replacing the increasing sequence by an almost increasing sequence. Now, we will prove the following theorem. Theorem 2.1. Let (Xn ) be an almost increasing sequence and the sequences (βn ) and (λn ) such that conditions (1.6), (1.7), (1.8), and (1.9) of Theorem 1.1 are satisfied. 295 ON ALMOST INCREASING SEQUENCES AND ITS APPLICATIONS If (pn ) is a sequence such that condition (1.11) of Theorem 1.2 is satisfied and m Pn δk−1 sn k = O Xm pn n=1 as m → ∞, (2.1) ∞ Pv δk 1 Pn δk−1 1 =O , pn Pn−1 pv Pv n=v+1 then the series (2.2) an λn is summable |N̄, pn ; δ|k for k ≥ 1 and 0 ≤ δ < 1/k. Remark 2.2. It may be noted that if we take (Xn ) as a positive nondecreasing sequence and δ = 0 in this theorem, then we get Theorem 1.2. In this case, condition (2.1) reduces to condition (1.12) and condition (2.2) reduces to 1 pn =O , P P P n n−1 v n=v+1 ∞ (2.3) which always holds. We need the following lemma for the proof of our theorem. Lemma 2.3 (see [5]). Under the conditions on (Xn ), (βn ), and (λn ) as taken in the statement of the theorem, the following conditions hold, when (1.9) is satisfied: nβn Xn = O(1) ∞ as n → ∞, (2.4) βn Xn < ∞. (2.5) n=1 Proof of Theorem 2.1. Let (Tn ) denotes the (N̄, pn ) mean of the series Then, by definition and changing the order of summation, we have Tn = n n v 1 1 pv ai λi = Pn − Pv−1 av λv . Pn v=0 P n v=0 i=0 an λn . (2.6) Then, for n ≥ 1, we have Tn − Tn−1 = n pn Pv−1 av λv . Pn Pn−1 v=1 (2.7) By Abel’s transformation, we have Tn − Tn−1 = n−1 pn pn ∆ Pv−1 λv sv + s n λn Pn Pn−1 v=1 Pn =− n−1 n−1 pn pn pn pv s v λ v + Pv sv ∆λv + sn λn Pn Pn−1 v=1 Pn Pn−1 v=1 Pn = Tn,1 + Tn,2 + Tn,3 , (2.8) 296 HIKMET S. ÖZARSLAN where Tn,i , i = 1, 2, 3, denotes the ith term in the sum. Since Tn,1 + Tn,2 + Tn,3 k ≤ 3k Tn,1 k + Tn,2 k + Tn,3 k , (2.9) to complete the proof of the theorem, it is enough to show that ∞ Pn δk+k−1 Tn,r k < ∞ for r = 1, 2, 3. p n n=1 (2.10) Now, when k > 1 applying Hölder’s inequality with indices k and k , where 1/k+1/k = 1, we have m+1 n=2 Pn pn δk+k−1 k m+1 Pn δk−1 −k n−1 Tn,1 k ≤ pv sv λv Pn−1 pn n=2 v=1 ≤ m+1 n=2 Pn pn δk−1 1 Pn−1 k−1 n−1 k k 1 pv sv λv pv Pn−1 v=1 v=1 n−1 = O(1) Pn δk−1 1 k k m+1 pv sv λv pn Pn−1 v=1 n=v+1 = O(1) m Pv δk−1 sv k λv λv k−1 pv v=1 = O(1) m Pv δk−1 sv k λv p v v=1 = O(1) m m−1 v ∆λv v=1 r =1 Pr pr δk−1 k sr m Pv δk−1 sv k + O(1)λm p v v=1 = O(1) m−1 ∆λv Xv + O(1)λm Xm v=1 = O(1) m−1 βv Xv + O(1)λm Xm v=1 = O(1) as m → ∞, (2.11) by virtue of the hypotheses of Theorem 1.2 and Lemma 2.3. Since vβv = O(1/Xv ) by (2.4), using the fact that Pv = O(vpv ) by (1.11), and |∆λn | ≤ βn by (1.6), and after applying Hölder’s inequality again, we have ON ALMOST INCREASING SEQUENCES AND ITS APPLICATIONS m+1 n=2 Pn pn δk+k−1 297 k m+1 Pn δk−1 −k n−1 Tn,2 k ≤ Pv ∆λv sv Pn−1 p n n=2 v=1 k m+1 Pn δk−1 −k n−1 = O(1) vpv βv sv Pn−1 pn n=2 v=1 k−1 n−1 m+1 Pn δk−1 1 n−1 k k 1 = O(1) pv vβv pv sv pn Pn−1 v=1 Pn−1 v=1 n=2 = O(1) m vβv v=1 = O(1) m vβv v=1 = O(1) m vβv k Pn δk−1 1 k m+1 pv sv pn Pn−1 n=v+1 k Pv δk−1 k sv pv k−1 vβv v=1 = O(1) m vβv v=1 = O(1) Pv δk−1 k sv pv Pv δk−1 k sv pv m−1 v ∆ vβv v=1 r =1 Pr pr δk−1 |sr |k m Pv δk−1 sv k + O(1)mβm pv v=1 = O(1) m−1 ∆ vβv Xv + O(1)mβm Xm v=1 = O(1) m−1 m−1 v ∆βv Xv + O(1) βv+1 Xv+1 + O(1)mβm Xm v=1 = O(1) v=1 as m → ∞, (2.12) by virtue of the hypotheses of Theorem 1.2 and Lemma 2.3. Finally, using the fact that Pv = O(vpv ) by (1.11), as in Tn,1 , we have m m Pn δk+k−1 Pn δk−1 Tn,3 k = O(1) sn k λn pn pn n=1 n=1 (2.13) = O(1) as m → ∞. Therefore, we get (2.10) and this completes the proof of the theorem. If we take pn = 1 for all values of n in this theorem, then we get a result concerning the |C, 1; δ|k summability factors. 298 HIKMET S. ÖZARSLAN References [1] [2] [3] [4] [5] [6] H. Bor, On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1, 147–149. MR 86d:40004. Zbl 554.40008. , On local property of |N̄, pn ; δ|k summability of factored Fourier series, J. Math. Anal. Appl. 179 (1993), no. 2, 646–649. MR 95a:42002. Zbl 797.42005. , A note on absolute summability factors, Int. J. Math. Math. Sci. 17 (1994), no. 3, 479–482. MR 95b:40007. Zbl 802.40004. G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 11,25a. Zbl 032.05801. S. M. Mazhar, A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica 25 (1997), no. 3, 233–242. MR 99a:40005. Zbl 885.40004. K. N. Mishra and R. S. L. Srivastava, On absolute Cesàro summability factors of infinite series, Portugal. Math. 42 (1983/84), no. 1, 53–61 (1985). MR 87a:40003. Zbl 597.40003. Hikmet S. Özarslan: Department of Mathematics, Erciyes University 38039, Kayseri, Turkey E-mail address: seyhan@erciyes.edu.tr