Characterizations Of (p, α)-convex Sequences

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1 Applied Mathematics E-Notes, , c ISSN Available free at mirror sites of ame/ Characterizatios Of p, α-covex Sequeces Xhevat Zahir Krasiqi Received 2 July 2016 Abstract The class of covex sequeces has importat applicatios i several braches of mathematics as well as their geeralizatios. I this paper, we have itroduced two ew classes of covex sequeces, the so-called p, α-covex sequeces ad p-starshaped sequeces. Moreover, the characterizatios of sequeces belogig to these class are discussed. 1 Itroductio The set of covex sequeces is oe of proper ad importat subset of the set of real sequeces. This class is raised as a result of efforts to solve several problems i mathematics. Sice the begiig of time, the sequeces that belog to that class, have cosiderable applicatios i some braches of mathematics, i particular i mathematical aalysis. For example, such sequeces are widely used i theory of iequalities see [12, 7, 8], i absolute summability of ifiite series see [1, 2], ad i theory of Fourier series, related to their uiform covergece ad the itegrability of their sum fuctios see as example [6], page 587. Let a =0 be a real sequece ad let the differeces of orders 0, 1, 2 of the sequece a =0 be defied by 0 a = a, 1 a = a 1 a, 2 a = a 2 2a 1 a, = 0, 1,..., ad throughout the paper we shall write a istead of 1 a. Next defiitio presets the well-kow otio of a covex sequece of order 2. if DEFINITION 1. A sequece a =0 is said to be covex of order 2 or just covex for all 0. 2 a 0, Various geeralizatios of covexity were studied by may authors. For istace, i [5] was itroduced ext: Mathematics Subject Classificatios: 26A51, 26A48, 26D15. Uiversity of Prishtia "Hasa Prishtia", Faculty of Educatio, Departmet of Mathematics ad Iformatics, Aveue "Mother Theresa " o. 5, Prishtië 10000, Kosovo 77

2 78 Characterizatios of p, α-covex Sequeces DEFINITION 2. umber p if A sequece a =0 is said to be p-covex for a positive real L p a 0, for all = 0, 1,..., where the differece operator L p is defied by L p a = a 2 1 pa 1 pa. Aother geeralizatio of the cocept of covexity ca be foud i [4] ad [3]. I [4] is give the followig defiitio: DEFINITION 3. If for a sequece a =0 the iequality a pa 1 0, holds true for every 0, the it is said that a =0 is a p-mootoe sequece. Here, we will say that a =0 is a p-icreasig sequece if the iequality holds true for every 0. a 1 pa 0, Two other classes of sequeces, the so-called, starshaped sequeces ad α-covex sequeces have bee itroduced i [9] ad [10]. Ideed, let α [0, 1]. DEFINITION 4. A sequece a =0 is called α-covex if the sequece is icreasig. αa 1 a 1 α a a 0 =1 DEFINITION 5. A sequece a =0 is called starshaped if a 1 a 0 a a 0 for 1. Let p be a real positive umber. Now, we itroduce two ew classes of sequeces as follows: DEFINITION 6. A sequece a =0 is called p-starshaped if a 1 a 0 p a a 0 for 1. 1

3 Xh. Z. Krasiqi 79 DEFINITION 7. A sequece a =0 is called p, α-covex if the sequece is p-icreasig. αa 1 a 1 α a a 0 =1 REMARK 8. We ote that: 1, α-covexity is the same with α-covexity, p, 1- covexity is the same with p-covexity, p, 0-covexity is the same with p-star-shapedess, 1, 1-covexity is the same with covexity, ad 1, 0-covexity is the same with starshapedess. REMARK 9. Note also that: 1-star-shapedess of a sequece is the same with its star-shapedess. Characterizig p, α-covex sequeces as well as p-starshaped sequeces, we are goig to accomplish the mai aim of this paper. 2 Mai Results We begi first with: THEOREM 10. The sequece a =0 is p, α-covex if ad oly if a1 a 0 αl p a 1 α a 0 0, for all {0, 1,... }. PROOF. The proof of this statemet is a immediate result of the Defiitio 1. The proof is complete. For p = 1 we obtai Corollary 11. COROLLARY 11 [10]. The sequece a =0 is α-covex if ad oly if α 2 a1 a 0 a 1 α a a 0 0, for all {0, 1,... }. THEOREM 12. The sequece a =0 is p, α-covex if ad oly if is a p-starshaped sequece. a a 0 α[a 1 a a a 0 ] =1,

4 80 Characterizatios of p, α-covex Sequeces PROOF. First let us write A := a a 0 α[a 1 a a a 0 ], {1, 2,... }, which ca be rewritte as A = αa 1 a 1 αa a 0, {1, 2,... }. The proof of this Lemma follows as a direct result of Lemma 2, ad the followig obvious equivaleces A 0 = 0 A 1 pa A 1 pa [αa 2 a 1 1 αa 1 a 0 ] p[αa 1 a 1 αa a 0 ] αl p a 1 α[a 1 a 0 pa 1 a 0 ] 0 a1 a 0 αl p a 1 α a 0 0. The proof is complete. For p = 1 we obtai Corollary 13. COROLLARY 13 [10]. The sequece a =0 is α-covex if ad oly if a a 0 α[a 1 a a a 0 ] =1, is a starshaped sequece. THEOREM 14. represeted by with 0, k 2. The sequece a =0 is p-starshaped if ad oly if it may be a = p 1 k p 1 1c 0, 2 PROOF. Our reasoig is similar to the proof of Lemma 3 i [11], page 3. Namely, let a 0 = c 0 ad a 1 = c 1 ad take = 2 i 1 we obtai a 2 2pc 1 2p 1c 0, which meas that there exists a umber c 2 0 such that a 2 = 2p c 1 c 2 2p 1c 0. 2

5 Xh. Z. Krasiqi 81 Now we assume that a = p 1 k p 1 1c 0. The for there exists c 1 p 0 so that we will have a 1 a 0 p a a 0 a 1 p a c 0 c 0 a 1 = c 1 p p 1 a 1 = p [ p 1 k [p 1c 0, k p 1 1c 0 c 0 ] which by mathematical iductio we obtai the represetatio 2. The proof is complete. COROLLARY 15. If the sequece a =0 is represeted by 2, the L p a = p [p 1 k 1 ] p 1 c 1 pc 2. c 0 Takig p = 1 i Theorem 2 ad Corollary 2 we get the followig: COROLLARY 16 [11]. The sequece a =0 is starshaped if ad oly if it may be represeted by a = k 1c 0 with 0, k 2. 3 COROLLARY 17 [11]. If the sequece a =0 is represeted by 3, the 2 a = c 2 c 1. THEOREM 18. represeted by The sequece a =0 is p, α-covex if ad oly if it may be a = p 1 k p 1 1c 0, 4

6 82 Characterizatios of p, α-covex Sequeces with 0 < p 1, ad c 0, 2. c 2 [ 1 1 p 1, 5 k 1 α ] 1 1 p 1 c 1, PROOF. O oe had, takig ito accout 4, we have αl p a = α[a 2 1 pa 1 pa ] = αp [p 1 k 1 O the other had, usig 4, we also have a1 a 0 1 α a 0 1 = 1 α p k [p 1]c 0 p k p pc 0 ] p 1 c 1 pc 2. 6 pc 0 c 0 = 1 αp c 1. 7 From 6 ad 7 we obtai a1 a 0 αl p a 1 α a 0 ] = p [αp p 1 k 1 α 1 c 1 αpc 2 1 αp c 1. Subsequetly, it follows that αl p a 1 α a1 a 0 a 0 0 if ad oly if 0 < p 1, k 1, ad c 2 The proof is complete. [ ] p α p 1 c 1.

7 Xh. Z. Krasiqi 83 COROLLARY 19 [10]. The sequece a =0 is α-covex if ad oly if it may be represeted by a = k 1c 0, with c c 1 ad c 0, 2. α THEOREM 20. If the sequece a =0 is p, α-covex, the it is p, β-covex for 0 β α ad 0 < p 1. PROOF. The proof follows from Theorem 2. Ideed, let the sequece a =0 be p, α-covex. The, it may be represeted by 4 with 5, [ ] c 2 1 p α p 1 c 1, ad c 0, 2. However, sice 0 β α the we also have [ ] c 2 1 p β p 1 c 1, with c 0, 2, which shows that the sequece a =0 is p, β-covex as well. The proof is complete. For p = 1, as a particular case, we obtai Corollary 21. COROLLARY 21 [10]. If the sequece a =0 is α-covex, the it is β-covex, for 0 β α. Refereces [1] H. Bor, A ew applicatio of covex sequeces, J. Class. Aal., 12012, [2] H. Bor ad Xh. Z. Krasiqi, A ote o absolute Cesào summability factors, Adv. Pure Appl. Math., 32012, [3] Xh. Z. Krasiqi, Some properties of p, q; r-covex sequeces, Appl. Math. E- Notes, , [4] L. M. Kocić, I. Z. Milovaović, A property of p, q-covex sequeces, Period. Math. Hugar., , [5] I. B. Lacković ad M. R. Jovaović, O a class of real sequeces which satisfy a differece iequality, Uiv. Beograd. Publ. Elektroteh. Fak. Ser. Mat., ,

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