ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES. 1. Introduction

Transcription

1 Joural of Classical Aalysis Volue 3, Nuber 2 208), doi:0.753/jca ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES AHMET KARAKAŞ Abstract. I the preset paper, a geeral theore dealig with A, p ;δ k suability ethod of ifiite series has bee proed by usig alost icreasig sequeces. Soe results hae also bee gie.. Itroductio Let a be a gie ifiite series with the partial sus s ).Letp ) be a sequece of positie ubers such that P = =0 p as ), P i = p i = 0, i ). ) Let A =a ) be a oral atrix, i.e., a lower triagular atrix of ozero diagoal etries. The A defies the sequece-to-sequece trasforatio, appig the sequece s =s ) to As =A s)), where A s)= =0 a s, = 0,,... 2) The series a is said to be suable A, p ;δ k, k adδ 0, if see [0]) where P = p ΔA s) k <, 3) ΔA s)=a s) A s). If we take δ = 0, the A, p ;δ k suability reduces to A, p k suability see [8]). If we take δ = 0, a = p P,theweget N, p k suability see [2]). Furtherore, if we take δ = 0, a = p P ad p = for all alues of, the A, p ;δ k suability reduces to C, k suability see [7]). Matheatics subject classificatio 200): 26D5, 40D5, 40F05, 40G99. Keywords ad phrases: Absolute atrix suability, alost icreasig sequeces, Hölder iequality, ifiite series, Mikowski iequality, suability factor. c D l,zagreb Paper JCA

2 34 A. KARAKAŞ 2. Kow result k I [3], Bor has proed the followig theore for N, p suability factors of ifiite series usig positie o-decreasig sequece. THEOREM. Let X ) be a positie o-decreasig sequece ad let there be sequeces β ) ad λ ) such that If Δλ β, 4) β 0 as, 5) = Δβ X <, 6) λ X = O). 7) s = OX ) as 8) ad p ) is a sequece such that the the series = a P λ p is suable N, p k,k. P = Op ), 9) P Δp = Op p + ), 0) REMARK. It should be oted that, fro the hypotheses of Theore, λ ) is bouded ad Δλ = O/) see [3]). 3. Mai result A positie sequece b ) is said to be alost icreasig if there exists a positie icreasig sequece c ) ad two positie costats K ad L such that Kc b Lc see []). May works o alost icreasig sequeces hae bee doe see [4] [6], [] [7]). The purpose of this paper is to geeralize Theore for A, p ;δ k suability. Before giig the ai theore, we ust first itroduce soe further otatios. Gie a oral atrix A =a ), we associate two lower seiatrices A =a ) ad  =â ) as follows: ad a = i= a i,, = 0,,... ) â 00 = a 00 = a 00, â = a a,, =,2,... 2)

3 ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES 35 It ay be oted that A ad  are the well-kow atrices of series-to-sequece ad series-to-series trasforatios, respectiely. The, we hae ad A s) = =0 ΔA s) = Now, we shall proe the followig theore. a s = i=0 i=0 a i a i 3) â i a i. 4) THEOREM 2. Let A =a ) be a positie oral atrix such that =+ =+ P p P a 0 =, = 0,,..., 5) a, a, for +, 6) ) p a = O, 7) P â,+ = O Δ â ) ), 8) ) δk ) P p Δ â = O as, 9) p p P ) δk ) P â,+ = O p as. 20) Let X ) be a alost icreasig sequece. If coditios 4) 7) ad 9) 0) of Theore ad P s k = OX ) as, 2) p are satisfied, the the series = a P λ p is suable A, p ;δ k,k ad 0 δ < /k. We should gie the followig leas for the proof of Theore 2. LEMMA. [8]) If X ) is a alost icreasig sequece, the uder the coditios 5) 6), we hae X β = O) as, 22) = β X <. 23) LEMMA 2. [9]) If the coditios 9) ad 0) are satisfied, the we hae ) ) P Δ = O. 24) p

4 36 A. KARAKAŞ 4. Proof of Theore 2 Let M ) deotes the A-trasfor of the series a λ P p ΔM = â a λ P p by 3) ad4). By applyig Abel s trasforatio, we get ) â λ P ΔM = Δ p a r + âp λ r= p = = Δ â λ P p ) P λ Δ â ) s + p P s + a P λ s p ) + â,+ λ Δ p = M, + M,2 + M,3 + M,4.. The, we hae a â,+ Δλ P + + )p + s s + a P λ s p To coplete the proof of Theore 2, by Mikowski s iequality, it is eough to show that P M,r k <, for r =,2,3,4. = p First, by applyig Hölder s iequality with idices k ad k,wherek > ad k + k =, we hae that P M, k ) δk+k ) } k P P =2 p =2 p Δ â ) λ s p ) δk+k ) } k P P =2 p Δ â ) λ k s k p k Δ â ) }. By )ad2), we hae Δ â )=â â,+ = a a, a,+ + a,+ = a a,. Thus usig ), 5)ad6) Δ â ) = a, a ) a.

5 ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES 37 Hece, =2 P p M, k =2 = O) = O) = O) P a k p P p P p P p = O) Δ λ ) k k λ k s k P p λ s k i= Pi p i P p =+ = O) Δλ X + O) λ X = O) β X + O) λ X = O) as, by irtue of the hypotheses of Theore 2 ad Lea. By usig 9) adhölder s iequality, we hae that P =2 p M,2 k = O) = O) = O) = O) = O) =2 =2 P p P p â,+ Δλ =2 P p P ) } k k Δ â ) λ k s k P p Δ â ) ) k k λ k λ s k s i k +O) λ i ) } δk+k k â,+ Δλ s ) } δk+k â,+ Δλ s k } k a k β s k =+ p P p β s k P p } â,+ β s k â,+ s k

6 38 A. KARAKAŞ = O) Δβ ) i= Pi = O) Δβ ) X + O)β X p i s i k +O)β i = O) Δβ X + O) β X + O)β X = O) as, by irtue of the hypotheses ) of Theore 2 ad Lea. Sice Δ P p = O ) by 24), as i M,,wehaethat =2 P p M,3 k = O) = O) P =2 p P =2 p = O) = O) = O) Δ â ) =2 P p P } k a k λ k λ s k p λ s k = O) as, by irtue of the hypothese of Theore 2 ad Lea. Fially, as i M,,wehaethat P p } k â,+ λ s â,+ λ k s k =+ â,+ λ k s k P p â,+ s k P = p M,4 k = O) = O) P = p P = p = O) as, p P λ s k by irtue of the hypotheses of Theore 2 ad Lea. ) k k P p ) k λ k λ s k

7 ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES Coclusios If we take X ) as a positie o-decreasig sequece, δ = 0ada = p P i Theore 2, the we get Theore. I this case, the coditio 2) reduces to the coditio 8). Also, the coditios 5) 20) are autoatically satisfied. Also, if we take δ = 0, a = p P ad p = for all alues of, the we get a result for C, k suability. REFERENCES [] N. K. BARI AND S. B. STEČKIN, Best approxiatios ad differetial properties of two cojugate fuctios, Trudy. Mosko. Mat. Obšč ), i Russia). [2] H. BOR, O two suability ethods, Math. Proc. Cabridge Philos. Soc ), [3] H. BOR, A ote o N, p k suability factors of ifiite series, Idia J. Pure Appl. Math ), [4] H. BOR, O absolute Riesz suability factors, Ad. Stud. Cotep. Math. Pusa), 3 2) 200), [5] H. BOR, A ote o absolute Riesz suability factors, Math. Iequal. Appl ), [6] H. BOR, A ew applicatio of alost icreasig sequeces, J. Coput. Aal. Appl ), [7] T. M. FLETT, O a extesio of absolute suability ad soe theores of Littlewood ad Paley, Proc. Lodo Math. Soc ), 3 4. [8] S. M. MAZHAR, A ote o absolute suability factors, Bull. Ist. Math. Acad. Siica ), [9] K.N.MISHRA AND R. S. L. SRIVASTAVA, O N, p suability factors of ifiite series, Idia J. Pure Appl. Math ), [0] H. S. ÖZARSLAN AND H. N. ÖĞDÜK, Geeralizatios of two theores o absolute suability ethods, Aust. J. Math. Aal. Appl. 2004), Article 3, 7 pp. [] H. S. ÖZARSLAN, A ew applicatio of alost icreasig sequeces, Miskolc Math. Notes ), [2] H. S. ÖZARSLAN, O geeralized absolute atrix suability, Asia Pacific J. Math. 2) 204), [3] H. S. ÖZARSLAN, A ew applicatio of absolute atrix suability, C. R. Acad. BulgareSci ), [4] H. S. ÖZARSLAN, A ew study o geeralized absolute atrix suability, Cou. Math. Appl ), [5] H. S. ÖZARSLAN, A ew applicatio of geeralized alost icreasig sequeces, Bull. Math. Aal. Appl ), 9 5. [6] H. S. ÖZARSLAN AND A. KARAKAŞ, A ew result o the alost icreasig sequeces, J. Cop. Aal. Appl ), [7] H. S. ÖZARSLAN AND B. KARTAL, A geeralizatio of a theore of Bor, J. Iequal. Appl ), 8. [8] W. T. SULAIMAN, Iclusio theores for absolute atrix suability ethods of a ifiite series. IV, Idia J. Pure Appl. Math. 34 ) 2003), Receied August 2, 208) Ahet Karakaş Departet of Matheatics Erciyes Uiersity Kayseri, Turkey e-ail: ahetkarakas985@hotail.co Joural of Classical Aalysis jca@ele-ath.co