In this paper a theorem on the absolute Nörlund summability
factors has been proved under more weaker conditions by using an almost increasing sequence

H. Bor

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Mathematical Communications 5(2000), 143-147 143On the absolute No¨rlund summability factorsHu¨seyin Bor∗Abstract. In this paper a theorem on the absolute No¨rlund sum-mability factors has been proved under more weaker conditions by usingan almost increasing sequence.Key words: No¨rlund summability, summability factors, almost in-creasing sequenceAMS subject classifications: 40D15, 40F05, 40G05Received July 6, 2000 Accepted October 12, 20001. IntroductionLet∑an be a given infinite series with the sequence of partial sums (sn) andwn = nan. By uαn and tαn we denote the n-th Cesa`ro means of order α, withα > − 1, of the sequences (sn) and (wn), respectively. The series∑an is said tobe summable | C,α |, if (see [4], [6])∞∑n=1| uαn − uαn−1 | =∞∑n=11n| tαn |<∞. (1)Let (pn) be a sequence of constants, real or complex, and let us writePn = p0 + p1 + p2 + ...+ pn = 0, (n ≥ 0) (2)The sequence-to-sequence transformationσn =1Pnn∑ν=0pn−νsv (3)defines the sequence (σn) of the No¨rlund mean of the sequence (sn), generated bythe sequence of coefficients (pn). The series∑an is said to be summable | N, pn |,if (see [7])∞∑n=1| σn − σn−1 | < ∞. (4)∗Department of Mathematics, Erciyes University, 38 039 Kayseri, Turkey, e-mail:bor@erciyes.edu.tr144 H.BorIn the special case wherepn =Γ(n+ α)Γ(α)Γ(n+ 1), α ≥ 0 (5)the No¨rlund mean reduces to the (C,α) mean and | N, pn | summability becomes| C,α | summability. For pn = 1 and Pn = n, we get the (C, 1) mean and then| N, pn | summability becomes | C, 1 | summability. For any sequence (λn) we write∆λn = λn − λn+1 and ∆2λn = ∆(∆λn) = ∆λn −∆λn+1.In [5] Kishore has proved the following theorem concerning | C, 1 | and | N, pn |summability methods.Theorem 1. Let p0 > 0, pn ≥ 0 and (pn) be a non-increasing sequence. If∑anis summable | C, 1 |, then the series ∑ anPn(n+ 1)−1 is summable | N, pn |.Ahmad [1] proved the following theorem for absolute No¨rlund summability factors.Theorem 2. Let (pn) be as in Theorem1. Ifn∑ν=11ν| tν |= O(Xn) as n→∞, (6)where (Xn) is a positive non-decreasing sequence and (λn) is a sequence such thatXnλn = O(1), (7)n∆Xn = O(Xn), (8)∑nXn | ∆2λn |<∞, (9)then the series∑anPnλn(n+ 1)−1 is summable | N, pn |.Later on Bor [3] has proved Theorem 2. under weaker conditions in the followingform.Theorem 3. Let (pn) be as in Theorem1. Let (Xn) be a positive non-decreasingsequence. If the conditions (6) and (7) of Theorem2. are satisfied and the sequences(λn) and (βn) are such that| ∆λn |≤ βn (10)βn → 0 (11)∑nXn | ∆βn |<∞, (12)then the series∑anPnλn(n+ 1)−1 is summable | N, pn |.On the absolute No¨rlund summability factors 1452. Main resultsThe aim of this paper is to prove Theorem 3 under more weaker conditions. For thiswe need the concept of an almost increasing sequence. A positive sequence (bn) issaid to be almost increasing if there exist a positive increasing sequence (cn) andtwo positive constants A and B such that Acn ≤ bn ≤ Bcn (see [2]). Obviously,every increasing sequence is almost increasing but the converse need not be true, ascan be seen from the example bn = ne(−1)n. So we are weakening the hypotheses ofthe theorem replacing the increasing sequence by any almost increasing sequence.Now we shall prove the following theorem.Theorem 4. Let (pn) be as in Theorem1 and let (Xn) be an almost increasingsequence. If the conditions (6), (7), (10) and (12) of Theorem2. and Theorem3.are satisfied, then the series∑anPnλn(n+ 1)−1 is summable | N, pn |.We need the following Lemma for the proof of our theorem.Lemma 1. Under the conditions on (Xn), (λn) and (βn), as taken in the statementof the theorem, the following conditions hold, when (12) is satisfied:nβnXn = O(1) as n→∞, (13)∞∑n=1βnXn <∞ (14)Proof. Let Acn ≤ bn ≤ Bcn, where (cn) is an increasing sequence. In this casenβnXn ≤ nBcn |∞∑ν=n∆βν |≤ nBcn∞∑ν=n| ∆βν |≤ B∞∑ν=nνcν | ∆βν |≤ (A/B)∞∑ν=nνXν | ∆βν |<∞.Hence, nβnXn = O(1) as n→∞.Again∞∑n=1βnXn ≤ B∞∑n=1cnβn = B∞∑n=1cn |∞∑ν=n∆βν |≤ B∞∑n=1cn∞∑ν=n| ∆βν |= B∞∑ν=1| ∆βν |ν∑n=1cn≤ B∞∑ν=1νcν | ∆βν |≤ (B/A)∞∑ν=1νXν | ∆βν |<∞.Thus∞∑n=1βnXn <∞. ✷Proof of Theorem 4. In order to prove the theorem, we need consider onlythe special case in which (N, pn) is (C, 1), that is, we shall prove that∑anλn is146 H.Borsummable | C, 1 |. Our theorem will then follow by means of Theorem 1. Let Tn bethe n-th (C, 1) mean of the sequence (nanλn), that is,Tn =1n+ 1n∑ν=1νaνλν (15)Using Abel’s transformation, we haveTn =1n+ 1n∑ν=1νaνλν =1n+ 1n∑ν=1∆λν(ν + 1)tν + λνtν= Tn,1 + Tn,2 , say.By (1), to completes the proof of the theorem, it is sufficient to show that∞∑n=1(1/n) | Tn,r |<∞ for r = 1, 2. (16)Now, we havem+1∑n=2(1/n) | Tn,1 | ≤m+1∑n=2(1/n(n+ 1)){n−1∑ν=1((ν + 1)/ν)ν | ∆λν || tν |}= O(1)m+1∑n=2(1/n2){n−1∑ν=1νβν | tν |}= O(1)m∑ν=1νβν | tν |m+1∑n=ν+11/n2 = O(1)m∑ν=1νβν | tν | /ν= O(1)m−1∑ν=1∆(νβν)ν∑r=1| tr | /r +O(1)mβmm∑ν=1| tν | /ν= O(1)m−1∑ν=1| ∆(νβν) | Xν +O(1)mβmXm= O(1)m−1∑ν=1|∆βν | νXν +O(1)m−1∑ν=1| βν+1 | Xν+1 +O(1)mβmXm= O(1) as m→∞,by (6), (10), (12), (13) and (14).Againm∑n=1(1/n) | Tn,2 | =m∑n=1| λn | (| tn | n)=m−1∑n=1∆ | λn |n∑ν=1| tν | ν+ | λm |m∑n=1| tn | n= O(1)m−1∑n=1| ∆λn | Xn +O(1) | λm | Xm= O(1)m−1∑n=1βnXn +O(1) | λm | Xm = O(1) as m→∞ ,by (6), (7), (10) and (14). This completes the proof of the theorem. ✷On the absolute No¨rlund summability factors 147References[1] Z.U.Ahmad, Absolute No¨rlund summability factors of power series andFourier series, Annales Polonici Math. 27(1972), 9–20.[2] S.Aljancˇic´, D.Arandelovic´, O-regularly varying functions, Publ. Inst.Math. 22(1977), 5–22.[3] H.Bor, Absolute No¨rlund summability factors of power series and Fourierseries, Annales Polonici Mathematici 56(1991), 11–17.[4] M.Feteke, Zur Theorie der divergenten Reihen, Math. es Termes Ertesito¨(Budapest) 29(1911), 719–726.[5] N.Kishore, On the absolute No¨rlund summability factors, Riv. Math. Univ.Parma 6(1965), 129–134.[6] E.Kogbentliantz, Sur le´s series absolument sommables par la me´thode desmoyennes arithme´tiques, Bull. Sci. Math. 49(1925), 234–256.[7] F.M.Mears, Some multiplication theorems for the No¨rlund mean, Bull. Amer.Math. Soc. 41(1935), 875–880.

On the absolute Nörlund summability factors

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