By applying the concept of a $beta$- power increasing sequence, the author presents a generalization of a result of Leindler [8] dealing with ${mid{bar{N},p_n}mid}_k$ summability for the $mid{bar{N},p_n,theta_{n}}mid_{k}$ summability factors. Some new results have also been obtained

Hüseyin Bor

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Mathematical Communications 13(2008), 265-270 265A new application of quasi power increasingsequencesHu¨seyi˙n Bor∗Abstract. By applying the concept of a β- power increasing se-quence, the author presents a generalization of a result of Leindler [8]dealing with | N¯, pn |k summability for the | N¯ , pn, θn |k summabilityfactors. Some new results have also been obtained.Key words: power increasing sequences, infinite series, absolutesummabilityAMS subject classifications: 40D15, 40F05, 40G99Received July 9, 2008 Accepted July 21, 20081. IntroductionA positive sequence (bn) is said to be almost increasing if there exists a positiveincreasing sequence (cn) and two positive constants A and B such that Acn ≤ bn ≤Bcn (see [1]). Obviously every increasing sequence is almost increasing. However,the converse need not be true as can be seen by taking the example, say bn =ne(−1)n. A positive sequence (γn) is said to be quasi β-power increasing sequenceif there exists a constant K = K(β, γ) ≥ 1 such thatKnβγn ≥ mβγm (1)holds for all n ≥ m ≥ 1 (see [8]). It should be noted that every almost increasingsequence is a quasi β-power increasing sequence for any nonnegative β, but theconverse need not be true as can be seen by taking the example, say γn = n−β forβ > 0. Let∑an be a given inﬁnite series with partial sums (sn). We denote by tnthe n-th (C,1) mean of the sequence (nan). A series∑an is said to be summable| C, 1 |k, k ≥ 1 , if (see [5],[7])∞∑n=11n| tn |k<∞. (2)∗Department of Mathematics, Erciyes University, 38 039 Kayseri, Turkey, e-mail:bor@erciyes.edu.tr266 H.BorLet (pn) be a sequence of positive numbers such thatPn =n∑v=0pv →∞ as n→∞, (P−i = p−i = 0, i ≥ 1). (3)The sequence-to-sequence transformationσn =1Pnn∑v=0pvsv (4)deﬁnes the sequence (σn) of the Riesz mean or simply the (N¯ , pn) mean of thesequence (sn), generated by the sequence of coeﬃcients (pn) (see [6]). The series∑an is said to be summable | N¯ , pn |k, k ≥ 1, if (see [2])∞∑n=1(Pn/pn)k−1 | ∆σn−1 |k<∞, (5)where∆σn−1 = − pnPnPn−1n∑v=1Pv−1av, n ≥ 1. (6)In the special case pn = 1 for all values of n | N¯ , pn |k summability is the same as| C, 1 |k summability.Let (θn) be any sequence of positive real constants. The series∑an is said to besummable | N¯, pn, θn |k, k ≥ 1, if (see [9])∞∑n=1θk−1n | ∆σn−1 |k<∞. (7)If we take θn = Pnpn , then | N¯ , pn, θn |k summability reduces to | N¯, pn |k summa-bility. Also if we take θn = n and pn = 1 for all values of n, then we get | C, 1 |ksummability. Furthermore if we take θn = n, then | N¯, pn, θn |k summability re-duces to | R, pn |k (see [3]) summability.2. Known resultsBor [ 4] has proved the following theorem for | N¯ , pn |k summability factors.Theorem A. Let (Xn) be an almost increasing sequence and let the conditionm∑n=11n| sn |k= O(Xm) (8)be satisfied. If the sequences (βn) and (λn) satisfy the conditions| ∆λn |≤ βn, (9)βn → 0, (10)∞∑n=1n | ∆βn | Xn <∞, (11)| λn | Xn = O(1), (12)Quasi power increasing sequences 267and furthermore if (pn) is a positive sequence such thatm∑n=1pnPn| sn |k= O(Xm), (13)then the series∑anλn is summable | N¯ , pn |k, k ≥ 1.Leindler [8] has proved Theorem A by using a quasi β− power increasing se-quence instead of an almost increasing sequence. His theorem is as follows:Theorem B. Let (Xn) be a quasi β− power increasing sequence for some 0 <β < 1. If all conditions from (8) to (13) are satisfied, then the series∑anλn issummable | N¯ , pn |k, k ≥ 1.3. The main resultThe aim of this paper is to generalize Theorem B for | N¯ , pn, θn |k summability.Now we shall prove the following theorem.Theorem. Let(θnpnPn)be a non-increasing sequence. If all the conditions ofTheorem B are satisfied with the condition (13) replaced bym∑n=1θk−1n(pnPn)k| sn |k= O(Xm), (14)then the series∑anλn is summable | N¯ , pn, θn |k, k ≥ 1.If we take θn = Pnpn , then we get Theorem B. In this case condition (14) reducesto condition (13) and the condition(θnpnPn)which is a non-increasing sequence isautomatically satisﬁed.We need the following lemma for the proof of our Theorem.Lemma 1 ([8]). Under the conditions on (Xn), (βn) and (λn) as taken in thestatement of the Theorem, the following conditions hold :nXnβn = O(1), (15)∞∑n=1βnXn <∞. (16)4. Proof of the TheoremLet (Tn) be the sequence of (N¯ , pn) mean of the series∑anλn. Then, by deﬁnition,we haveTn =1Pnn∑v=0pvv∑r=1arλr. (17)ThenTn − Tn−1 = pnPnPn−1n∑v=1Pv−1avλv, n ≥ 1. (18)268 H.BorUsing Abel’s transformation, we getTn − Tn−1 = − pnPnPn−1n−1∑v=1pvsvλv +pnPnPn−1n−1∑v=1Pvsv∆λv +pnPnsnλn= Tn,1 + Tn,2 + Tn,3, say.To prove the theorem, by Minkowski’s inequality, it is suﬃcient to show that∞∑n=1θk−1n | Tn,r |k<∞, for r = 1, 2, 3. (19)Firstly by using Ho¨lder’s inequality, we have thatm+1∑n=2θk−1n | Tn,1 |k =m+1∑n=2θk−1n |pnPnPn−1n−1∑v=1pvsvλv |k≤m+1∑n=2θk−1n(pnPn)k 1Pn−1n−1∑v=1pv | λv || λv |k−1| sv |k×{1Pn−1n−1∑v=1pv}k−1= O(1)m∑v=1pv | λv || sv |km+1∑n=v+1(θnpnPn)k−1pnPnPn−1= O(1)m∑v=1pv | λv || sv |k(θvpvPv)k−1 m+1∑n=v+1pnPnPn−1= O(1)m∑v=1pv | λv || sv |k(θvpvPv)k−1 1Pv= O(1)m∑v=1| λv | θk−1v(pvPv)k| sv |k= O(1)m−1∑v=1∆ | λv |v∑r=1θk−1r(prPr)k| sr |k+O(1) | λm |m∑v=1θk−1v(pvPv)k| sv |k= O(1)m−1∑v=1| ∆λv | Xv +O(1) | λm | Xm= O(1)m−1∑v=1βvXv +O(1) | λm | Xm = O(1) as m→∞,Quasi power increasing sequences 269in view of hypotheses of the Theorem and Lemma.Also we get thatm+1∑n=2θk−1n | Tn,2 |k=m+1∑n=2θk−1n |pnPnPn−1n−1∑v=1Pvsv∆λv |k≤m+1∑n=2θk−1n(pnPn)k 1Pn−1n−1∑v=1Pv | sv |k βv ×{1Pn−1n−1∑v=1Pvβv}k−1= O(1)m∑v=1Pv | sv |k βvm+1∑n=v+1(θnpnPn)k−1pnPnPn−1= O(1)m∑v=1Pvβv | sv |k(θvpvPv)k−1 m+1∑n=v+1pnPnPn−1= O(1)m∑v=1βv | sv |k(θvpvPv)k−1= O(1)(θ1p1P1)k−1 m∑v=1βv | sv |k= O(1)m∑v=1vβv| sv |kv= O(1)m−1∑v=1∆(vβv)v∑r=1| sr |kr+O(1)mβmm∑v=1| sv |kv= O(1)m−1∑v=1| ∆(vβv) | Xv +O(1)mβmXm= O(1)m−1∑v=1| (v + 1) | ∆βv | − | βv | | Xv +O(1)mβmXm= O(1)m−1∑v=1v | ∆βv | Xv +O(1)m−1∑v=1| βv | Xv +O(1)mβmXm= O(1) as m→∞,by virtue of hypotheses of the Theorem and Lemma..Finally, as in Tn,1 we have thatm∑n=1θk−1n | Tn,3 |k =m∑n=1θk−1n |pnPnsnλn |k= O(1)m∑n=1θk−1n(pnPn)k| λn || λn |k−1| sn |k= O(1)m∑n=1| λn | θk−1n(pnPn)| sn |k= O(1) as m→∞.270 H.BorTherefore we get thatm∑n=1θk−1n | Tn,r |k= O(1) as m→∞, for r = 1, 2, 3.This completes the proof of the Theorem. ✷If we take pn = 1 for all values of n and θn = n, then we get a new resultconcerning the | C, 1 |k summability factors. Also, if we take pn = 1 for all values ofn, then we have a new result for | C, 1, θn |k summability. Furthermore, if we takeθn = n, then we have another new result for | R, pn |k summability. Finally, if wetake pn = 1n+1 , then we get a result for | N¯, 1n+1 , θn |k summability.References[1] S. Aljancˇic´, D.Arandelovic´, O-regularly varying functions, Publ. Inst.Math. 22(1977), 5-22.[2] H.Bor, A note on two summability methods, Proc. Amer. Math. Soc.98(1986), 81-84.[3] H.Bor, On the relative strength of two absolute summability methods, Proc.Amer. Math. Soc. 113(1991), 1009-1012.[4] H.Bor, An application of almost increasing sequences, Internat. J. Math .Math. Sci. 23(2000), 859-863.[5] T.M.Flett, On an extension of absolute summability and some theorems ofLittlewood and Paley, Proc. London Math. Soc. 7(1957), 113-141.[6] G.H.Hardy, Divergent Series, Oxford Univ. Press., Oxford, 1949.[7] E.Kogbetliantz, Sur les se´ries absolument par la me´thode des moyannesarithme´tiques, Bull. Sci. Math. 49(1925), 234-256.[8] L.Leindler, A new application of quasi power increasing sequences,Publ.Math.Debrecen 58 (2001), 791-796.[9] W.T. Sulaiman, On some summability factors of infinite series, Proc. Amer.Math. Soc. 115(1992), 313-317.

A new application of quasi power increasing sequences

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