Hüseyin Bor

Fixed Point Theory and Applications

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A new application of quasi power increasing sequences. II

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Bor Fixed Point Theory and Applications 2013, 2013:75http://www.ﬁxedpointtheoryandapplications.com/content/2013/1/75RESEARCH Open AccessA new application of quasi power increasingsequences. IIHüseyin Bor**Correspondence:hbor33@gmail.comP.O. Box 121, Bahçelievler, Ankara,TR-06502, TurkeyAbstractIn this paper, we prove a general theorem dealing with absolute Cesàro summabilityfactors of inﬁnite series by using a quasi-f -power increasing sequence instead of aquasi-σ -power increasing sequence. This theorem also includes several new results.MSC: 26D15; 40D15; 40F05; 40G99; 46A45Keywords: absolute summability; increasing sequences; sequence spaces; Hölderinequality; Minkowski inequality; inﬁnite series1 IntroductionA positive sequence (bn) is said to be almost increasing if there exists a positive increas-ing sequence cn and two positive constants A and B such that Acn ≤ bn ≤ Bcn (see []).A sequence (λn) is said to be of bounded variation, denoted by (λn) ∈ BV , if∑∞n= |λn| =∑∞n= |λn – λn+| <∞. A positive sequence X = (Xn) is said to be a quasi-σ -power increas-ing sequence if there exists a constant K = K(σ ,X) ≥  such that KnσXn ≥ mσXm holdsfor all n ≥ m ≥  (see []). It should be noted that every almost increasing sequence is aquasi-σ -power increasing sequence for any nonnegative σ , but the converse may not betrue as can be seen by taking an example, say Xn = n–σ for σ > . Let (ϕn) be a sequence ofcomplex numbers and let∑an be a given inﬁnite series with partial sums (sn). We denoteby zαn and tαn the nth Cesàro means of order α, with α > –, of the sequences (sn) and (nan),respectively, that is,zαn =Aαnn∑v=Aα–n–vsv, ()tαn =Aαnn∑v=Aα–n–vvav, ()whereAαn =(n + αn)= (α + )(α + ) · · · (α + n)n! =O(nα), Aα–n =  for n > . ()The series∑an is said to be summable ϕ – |C,α|k , k ≥  and α > –, if (see [, ])∞∑n=∣∣ϕn(zαn – zαn–)∣∣k =∞∑n=n–k∣∣ϕntαn ∣∣k <∞. ()© 2013 Bor; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.Bor Fixed Point Theory and Applications 2013, 2013:75 Page 2 of 5http://www.ﬁxedpointtheoryandapplications.com/content/2013/1/75In the special case if we take ϕn = n–k , then ϕ – |C,α|k summability is the same as |C,α|ksummability (see []). Also, if we take ϕn = nδ+–k , then ϕ – |C,α|k summability reduces to|C,α; δ|k summability (see []).2 The known resultsTheorem A ([]) Let (λn) ∈ BV and let (Xn) be a quasi-σ -power increasing sequence forsome σ ( < σ < ). Suppose also that there exist sequences (βn) and (λn) such that|λn| ≤ βn, ()βn →  as n→ ∞, ()∞∑n=n|βn|Xn <∞, ()|λn|Xn =O() as n→ ∞. ()If there exists an 	 >  such that the sequence (n	–k|ϕn|k) is nonincreasing and if the sequence(wαn) deﬁned by (see [])wαn =⎧⎨⎩|tαn |, α = ,max≤v≤n |tαv |,  < α < ,()satisﬁes the conditionm∑n=(|ϕn|wαn)knk =O(Xm) as m→ ∞, ()then the series∑anλn is summable ϕ – |C,α|k , k ≥ ,  < α ≤  and kα + 	 > .Remark  Here, in the hypothesis of TheoremA, we have added the condition ‘(λn) ∈ BV ’because it is necessary.TheoremB ([]) Let (Xn) be a quasi-σ -power increasing sequence for some σ ( < σ < ). Ifthere exists an 	 >  such that the sequence (n	–k|ϕn|k) is nonincreasing and if the conditionsfrom () to () are satisﬁed and if the conditionm∑n=(|ϕn|wαn)knkXk–n=O(Xm) as m→ ∞, ()is satisﬁed, then the series∑anλn is summable ϕ – |C,α|k , k ≥ ,  < α ≤  and k(α – ) +	 > .Remark  It should be noted that condition () is the same as condition () when k = .When k > , condition () is weaker than condition () but the converse is not true. Asin [], we can show that if () is satisﬁed, then we getm∑n=(|ϕn|wαn)knkXk–n=O( Xk–) m∑n=(|ϕn|wαn)knk =O(Xm).Bor Fixed Point Theory and Applications 2013, 2013:75 Page 3 of 5http://www.ﬁxedpointtheoryandapplications.com/content/2013/1/75If () is satisﬁed, then for k >  we obtain thatm∑n=(|ϕn|wαn)knk =m∑n=Xk–n(|ϕn|wαn)knkXk–n=O(Xk–m) m∑n=(|ϕn|wαn)knkXk–n=O(Xkm) =O(Xm).Also, it should be noted that the condition ‘(λn) ∈ BV ’ has been removed.3 Themain resultThe aim of this paper is to extend Theorem B by using a general class of quasi powerincreasing sequence instead of a quasi-σ -power increasing sequences. For this purpose,we need the concept of quasi-f -power increasing sequence. A positive sequence X = (Xn)is said to be a quasi-f -power increasing sequence, if there exists a constant K = K(X, f )such that KfnXn ≥ fmXm, holds for n≥m≥ , where f = (fn) = [nσ (logn)η,η ≥ , < σ < ](see []). It should be noted that if we take η = , then we get a quasi-σ -power increasingsequence. Now, we will prove the following theorem.Theorem Let (Xn) be a quasi-f -power increasing sequence. If there exists an 	 >  suchthat the sequence (n	–k|ϕn|k) is non-increasing and if the conditions from () to () and() are satisﬁed, then the series∑anλn is summable ϕ – |C,α|k , k ≥ ,  < α ≤  andk(α – ) + 	 > .We need the following lemmas for the proof of our theorem.Lemma  ([]) If  < α ≤  and ≤ v≤ n, then∣∣∣∣∣v∑p=Aα–n–pap∣∣∣∣∣≤ max≤m≤v∣∣∣∣∣m∑p=Aα–m–pap∣∣∣∣∣. ()Lemma  ([]) Under the conditions on (Xn), (βn), and (λn) as expressed in the statementof the theorem, we have the following:nβnXn =O() as n→ ∞, ()∞∑n=βnXn <∞. ()4 Proof of the theoremLet (Tαn ) be the nth (C,α), with  < α ≤ , mean of the sequence (nanλn). Then, by (), wehaveTαn =Aαnn∑v=Aα–n–vvavλv. ()First, applying Abel’s transformation and then using Lemma , we get thatTαn =Aαnn–∑v=λvv∑p=Aα–n–ppap +λnAαnn∑v=Aα–n–vvav,Bor Fixed Point Theory and Applications 2013, 2013:75 Page 4 of 5http://www.ﬁxedpointtheoryandapplications.com/content/2013/1/75∣∣Tαn ∣∣≤ Aαnn–∑v=|λv|∣∣∣∣∣v∑p=Aα–n–ppap∣∣∣∣∣ + |λn|Aαn∣∣∣∣∣n∑v=Aα–n–vvav∣∣∣∣∣≤ Aαnn–∑v=Aαv wαv |λv| + |λn|wαn= Tαn, + Tαn,.To complete the proof of the theorem, by Minkowski’s inequality, it is suﬃcient to showthat∞∑n=n–k∣∣ϕnTαn,r∣∣k <∞ for r = , .Now, when k > , applying Hölder’s inequality with indices k and k′, where k +k′ = , weget thatm+∑n=n–k∣∣ϕnTαn,∣∣k ≤m+∑n=n–k(Aαn)–k|ϕn|k{ n–∑v=Aαv wαv |λv|}k≤m+∑n=n–kn–αk|ϕn|kn–∑v=vαk(wαv)k|λv|k ×{ n–∑v=}k–= O()m∑v=vαk(wαv)k(βv)k m+∑n=v+n	–k|ϕn|knk(α–)+	+= O()m∑v=vαk(wαv)k(βv)kv	–k|ϕv|k m+∑n=v+nk(α–)+	+= O()m∑v=vαk(wαv)k(βv)kv	–k|ϕv|k∫ ∞vdxxk(α–)+	+= O()m∑v=βv(βv)k–(wαv |ϕv|)k= O()m∑v=βv( vXv)k–(wαv |ϕv|)k= O()m–∑v=(vβv)v∑r=(|ϕr|wαr )krkXk–r+O()mβmm∑v=(|ϕv|wαv )kvkXk–v= O()m–∑v=∣∣(vβv)∣∣Xv +O()mβmXm= O()m–∑v=∣∣(v + )βv – βv∣∣Xv +O()mβmXm= O()m–∑v=v|βv|Xv +O()m–∑v=βvXv +O()mβmXm= O() asm→ ∞,Bor Fixed Point Theory and Applications 2013, 2013:75 Page 5 of 5http://www.ﬁxedpointtheoryandapplications.com/content/2013/1/75by virtue of the hypotheses of the theorem and Lemma . Finally, we have thatm∑n=n–k∣∣ϕnTαn,∣∣k =m∑n=|λn||λn|k–n–k(wαn |ϕn|)k= O()m∑n=|λn|( Xn)k–n–k(wαn |ϕn|)k= O()m–∑n=|λn|n∑v=(|ϕv|wαv )kvkXk–v+O()|λm|m∑n=(|ϕn|wαn)knkXk–n= O()m–∑n=|λn|Xn +O()|λm|Xm= O()m–∑n=βnXn +O()|λm|Xm =O() asm→ ∞,by virtue of the hypotheses of the theorem and Lemma . This completes the proof of thetheorem. If we take 	 =  and ϕn = n–k (resp. 	 = , α =  and ϕn = n–k ), then we get anew result dealing with |C,α|k (resp. |C, |k) summability factors of inﬁnite series. Also,if we take 	 =  and ϕn = nδ+–k , then we get another new result concerning the |C,α; δ|ksummability factors of inﬁnite series. Furthermore, if we take (Xn) as an almost increasingsequence, then we get the result of Bor and Seyhan under weaker conditions (see []).Finally, if we take η = , then we obtain Theorem B.Competing interestsThe author declares that he has no competing interests.AcknowledgementsDedicated to Professor Hari M. Srivastava.The author express his thanks the referees for their useful comments and suggestions.Received: 24 December 2012 Accepted: 18 March 2013 Published: 28 March 2013References1. Bari, NK, Stecˇkin, SB: Best approximation and diﬀerential properties of two conjugate functions. Tr. Mosk. Mat. Obsˆ. 5,483-522 (1956) (in Russian)2. Leindler, L: A new application of quasi power increasing sequences. Publ. Math. (Debr.) 58, 791-796 (2001)3. Balcı, M: Absolute ϕ-summability factors. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 29, 63-80 (1980)4. Kogbetliantz, E: Sur lés series absolument sommables par la méthode des moyennes arithmétiques. Bull. Sci. Math.49, 234-256 (1925)5. Flett, TM: On an extension of absolute summability and some theorems of Littlewood and Paley. Proc. Lond. Math.Soc. 7, 113-141 (1957)6. Flett, TM: Some more theorems concerning the absolute summability of Fourier series. Proc. Lond. Math. Soc. 8,357-387 (1958)7. Bor, H, Özarslan, HS: A study on quasi power increasing sequences. Rocky Mt. J. Math. 38, 801-807 (2008)8. Pati, T: The summability factors of inﬁnite series. Duke Math. J. 21, 271-284 (1954)9. Bor, H: A new application of quasi power increasing sequences. I. J. Inequal. Appl. 2013, 69 (2013)10. Sulaiman, WT: A note on |A|k summability factors of inﬁnite series. Appl. Math. Comput. 216(9), 2645-2648 (2010)11. Sulaiman, WT: Extension on absolute summability factors of inﬁnite series. J. Math. Anal. Appl. 322, 1224-1230 (2006)12. Bosanquet, LS: A mean value theorem. J. Lond. Math. Soc. 16, 146-148 (1941)13. Bor, H, Seyhan, H: A note on almost increasing sequences. Ann. Soc. Math. Pol., 1 Comment. Math. 39, 37-42 (1999)doi:10.1186/1687-1812-2013-75Cite this article as: Bor: A new application of quasi power increasing sequences. II. Fixed Point Theory and Applications2013 2013:75.

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A new application of quasi power increasing sequences. II

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