A study on generalized absolute summability factors for a triangular matrix

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Ποδαργη Βασιλείου
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Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 doi: 0.376/proc.20.2.06 Available olie at www.eap.

edu.tr Received 3 August 2009, revised 26 March 200, accepted 5 April 200 Abstract. I this paper, we establish a suability factor theore for suability A,δ.

1 Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 doi: 0.376/proc Available olie at A study o geeralized absolute suability factors for a triagular atrix Ere Savaş Departet of Matheatics, Istabul Ticaret Uiversity, Üsüdar-Istabul, Turey; eresavas@yahoo.co; esavas@iticu.edu.tr Received 3 August 2009, revised 26 March 200, accepted 5 April 200 Abstract. I this paper, we establish a suability factor theore for suability A,δ. This paper is a extesio of the ai result of Savas, E. A study o absolute suability factors for a triagular atrix. Math. Ieq. Appl., 2009, 2(9), Key words: absolute suability, suability factors, triagular atrix.. INTRODUCTION Savas 2] obtaied sufficiet coditios for a λ to be A -suable, N. I this paper, a theore o A,δ -suability ethods is proved. This theore icludes a ow result. Let A be a lower triagular atrix, {s } a sequece. The A : A series a is said to be suable A, if 0 a s. ad it is said to be suable A,δ, ad δ 0 if (see, ]) A A <, () δ+ A A <. (2) We ay associate with A two lower triagular atrices A ad Â defied as follows: ad ā a r,, 0,,2,..., r â ā ā,,,2,3,...

2 6 Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 Also we shall defie y : i0 a i s i 0λ a i i a i i0 0 a i 0 λ a ā λ a ad Y : y y 0 (ā ā, )λ a 0 â λ a. (3) Give ay sequece {x }, the otatio x O() eas x ad /x. For ay atrix etry a, a : a a MAIN RESULT Theore. Let A be a lower triagular atrix with oegative etries such that (i) ā 0, 0,,..., (ii) a, a for +, (iii) a O(), ad (iv) a â,+ O(a ). ) (v) δ â O ( δ a + (vi) (vii) (viii) ( δ â + O δ). + ad Let t deote the th (C,) ea of {a }. If (ix) δ a λ t, δ a λ t, δ a λ + t, the the series a λ is suable A,δ,,0 δ < /. Proof. Fro (i) it follows that â,0 0. Also (â λ )a is bouded. Usig (3) we ay write, for exaple:

3 E. Savaş: Geeralized absolute suability factors for a triagular atrix 7 Y + (â λ ) a (â λ ) r ra r r ra r ] (â λ ) ra r + âλ r a + ( â )λ t + â,+ ( λ ) + t â,+ λ + t + ( + )a λ t T + T 2 + T 3 + T 4. I order to prove our theore it is sufficiet, by Miowsi s iequality, to show that δ+ T r <, for r,2,3,4. Usig Hölder s iequality, (iii), (v), ad (vii) I : δ+ T δ+ ( δ+ â λ t δ+ ( δ (a ). â λ + ) )( ) â λ t â λ t â λ t δ (a ) â + λ t δ â + δ a λ t t

4 8 Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 Usig Hölder s iequality, (iii), (iv), (vi), ad (viii) I 2 : δ+ T 2 δ+ δ+ â,+ λ + t δ+ λ t â,+ â,+ ( λ ) + δ+ λ t a â,+ δ (a ). ] ] ] a λ t â,+ a λ t δ (a ) â,+ + a λ t δ â,+ + δ a λ t Usig Hölder s iequality, (iii), (iv), (vi), ad (ix) δ+ T 3 δ+ â,+ λ + δ+ a λ + â,+ t δ+ λ + a t â,+ δ (a ). t t a â,+ ] ] λ + a t â,+ λ + a t δ â,+ + δ a λ + t ] a â,+ ]

5 E. Savaş: Geeralized absolute suability factors for a triagular atrix 9 Fially, usig (iii) ad (v) δ+ T 4 δ+ ( + )a λ t δ (a ) a λ t δ a λ t Settig δ 0 i the theore yields the followig corollary:. Corollary (2]). Let A be a triagle satisfyig coditios (i) (iv) of Theore ad let t deote the th (C,) ea of {a }. If (v) (vi) (vii) a λ t, a λ t, a λ + t, the the series a λ is suable A, N. Rear. I ust ote that i the theore of Savas 2], the followig coditio should be added. (vii) a λ + t. 3. CONCLUSION Let a v deote a series with partial sus s. For a ifiite atrix T, the th ter of the T -trasfor of {s } is deoted by t v0 Let σ α deote the th ters of the trasfor of a Cesáro atrix (C,α) of a sequece {s }. I 957 Fleet ] gave the followig defiitio. A series a, with partial sus s, is said to be absolutely (C,α) suable of order, writte a is suable C,α, if t v s v. σ α σ α <. (4) Recetly, Savas 2] obtaied a absolute suability factor theore for lower triagular atrices. A suability factor theore for suability A,δ has ot bee studied so far. The preset paper fills up a gap i the existig literature. ACKNOWLEDGEMENTS I wish to tha the referees for their careful readig of the auscript ad for their helpful suggestios.

6 20 Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 REFERENCES. Fleet, T. M. O a extesio of absolute suability ad soe theores of Littlewood ad Paley. Proc. Lodo Math. Soc., 957, 3(7), Savas, E. A study o absolute suability factors for a triagular atrix. Math. Ieq. Appl., 2009, 2(9), Kolursete aatriseetluste üldistatud absoluutse sueeruvuse teguritest Ere Savaş Olgu A olure aatris ja δ 0. Artilis o defieeritud aatriseetlusega A -järu üldistatud absoluutse sueeruvuse eh A,δ -sueeruvuse õiste ja leitud piisavad tigiused selles, et rida Σλ a oles A,δ -sueeruv, st et arvud λ olesid eetluse A -järu üldistatud absoluutse sueeruvuse tegurites. Saadud tuleus üldistab autori varaseat tuleust (vt Savas, E. Math. Ieq. Appl., 2009, 2(9), 4 46).

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