Advaces i Dyamical Systems ad Applicatios. ISSN 0973-532 Volume Number 2006, pp. 79 89 c Research Idia Publicatios http://www.ripublicatio.com/adsa.htm O Summability Factors for N, p B.E. Rhoades Departmet of Mathematics, Idiaa Uiversity, Bloomigto, IN 47405-706, U.S.A. E-mail: rhoades@idiaa.edu Erem Savaş Departmet of Mathematics, Yüzücü Yil Uiversity, Va, Turey E-mail: eremsavas@yahoo.com Abstract I this paper ecessary ad sufficiet coditios are obtaied for a λ to be N, summable, >, wheever a is N, p, q bouded. AMS subject classificatio: 40F05, 40D25. Keywords: Absolute summability factors, ifiite series, weighted mea.. Itroductio Let a be a give ifiite series with partial sums s. Let {t } deote the sequece of N, p, q meas of {s }. The N, p, q trasform of a sequece {s } is defied by where R := t = R p ν s ν, p ν 0 for ay p = q = R = 0. Necessary ad sufficiet coditios for the N, p, q method to be regular are Received March 20, 2006; Accepted July 2, 2006
80 B.E. Rhoades ad Erem Savaş i lim p ν /R = 0 for each ν ad ii p ν < H R, where H is a positive costat idepedet of. Let {T } deote the sequece of N, meas of a sequece {s }. The T = s ν 0. Defie the sequece of costats {c } formally by meas of the idetity p x = c x, c i = 0, i. We also write c = c 0 + c + + c. We deote by M the set of sequeces {p } satisfyig p > 0, p + p p +2 p +, = 0,,.... Let {p }, { } be positive sequeces. A series a is said to be N, p, q bouded, or a N, p, q if p ν s ν = OR as. ν= If X ad Y are ay two methods of summability, we say that {λ } belogs to the class [X, Y ] if a λ is Y summable wheever a is X summable. I 966, Das [2] proved the followig theorem. Theorem.. Let {p } M, N, summable. 0. The if a is N, p, q summable, it is Recetly Sigh ad Sharma [5] proved the followig theorem. Theorem.2. Let {p } M, > 0 ad let { } be a mootoic odecreasig sequece for 0. The a ecessary ad sufficiet coditio that a λ is N,
Summability Factors 8 summable wheever is that a N, p, q, λ <, Q λ <, ad + + 2 λ <, s λ <. The aim of this paper is to geeralize the above theorem for N, summability methods,. We shall prove the followig theorem. Theorem.3. Let {p } M, > 0, { } be mootoe oicreasig ad = O. A ecessary ad sufficiet coditio that λ N, p, q, N, wheever i a [N, p, q], ii iii iv λ <, Q λ <, Q+ Q λ <, + v is that vi Q+ + Q 2 λ <, q s λ <. This theorem geeralizes several ow results of Mahzar [3], Daiel [], Sulaima [6], ad others.
82 B.E. Rhoades ad Erem Savaş 2. Lemmas We eed the followig lemmas for the proof of our theorem. Lemma 2.. [] Let p M. The c 0 > 0, c 0, =, 2,..., 2 3 4 c x is absolutely coverget for x, ad c > 0, except whe p =, i which case c = 0. Lemma 2.2. [2] If the Lemma 2.3. [2] We have t := R s = p ν s ν, c ν R ν s ν. c µr µ =, where c, R, ad are as defied as i Sectio. 3. Proof of the Theorem Proof. Let t be as i Lemma 2.2 ad let T deote the N, mea of the series a λ. The, by defiitio, we have T = a r λ r = a ν λ ν. r=0 The, for, we have T T = Q ν a ν λ ν.
Summability Factors 83 Usig summatio by parts, [ ] ν T T = Q ν λ ν a r a r r=0 r=0 [ = Q ν λ ν a r Q ν λ ν+ = [ r=0 Q ν λ ν ] a r r=0 ] a r + s λ r=0 = q s ν λ ν + Q ν s ν λ ν + s λ = T + T 2 + T 3, say. Usig Miowsi s iequality, it will be sufficiet to show that sice T 3 is coditio vi: T i <, i =, 2, T = q λ ν s ν = q λ ν = q λ ν c c = q c λ µ [ = q c λ ν + c µλ ]
84 B.E. Rhoades ad Erem Savaş = q = T,, + T,,2, say. c λ ν λ µ c µ Sice = O ad t, usig Lemmas 2., 2.2, 2.3 ad coditio iii, we have T,, = q q R Q µ t µ ν= λ ν λ ν Q ν ν= λ ν Q νq ν λ ν Q νq ν λ ν Qν Q ν =ν+ Qν λ ν. c λ ν c λ ν, c R µ
Summability Factors 85 Usig coditio ii, T,,2 = T 2 = q q q λ c µ λ c Q µr µ λ c µr µ Q λ Q λ, = Q ν λ ν s ν Q ν λ ν c = Q ν λ ν c = q c + + + + λ ν c Q ν+ c Q ν+ 2 λ ν + c µ λ = T,2, + T,2,2 + T,2,3 + T,2,4, say. λ ν
86 B.E. Rhoades ad Erem Savaş Usig coditio iii ad applyig Hölder s iequality, we have T,2, = q q c + λ ν R µ qν+ qν+ q ν qν+ q ν qν+ q ν Qν + λ ν + λ ν Q νqν λ ν Q ν c + c + λ ν λ ν Q ν R µ c λ ν Q νqν q ν =ν+ Q ν Qν λ ν λ ν. λ ν
Summability Factors 87 Usig iv ad Hölder s iequality, T,2,2 = q q Q Q ν c Q ν+ λ ν + Q ν+ + λ ν c Q ν+ λ ν Q ν Q ν+ + λ ν R µ c Qν Q ν+ qν λ ν + Qν Q ν+ q + qν ν λ ν =ν+ Qν+ + Q ν q qν ν λ ν Qν+ Qν λ ν. +
88 B.E. Rhoades ad Erem Savaş Usig Hölder s iequality ad v, T,2,3 = q q q Q Qν+ c Q ν+ + 2 λ ν Qν+ + Q ν+ 2 λ ν + 2 λ ν + c Q ν+ Q ν+ + Q ν Q ν 2 λ ν + Q R µ c νqν 2 λ ν Qν 2 λ ν. Usig iv, T,2,4 = λν λν R ν t ν c µ R ν t ν c λν Q µ
Summability Factors 89 λ ν q Q λ Q Q λ. The proof is complete. Refereces [] E.C. Daiel. O the N, p summability of ifiite series, J. Math. Jabalpur, 2:39 48, 966. [2] G. Das. O some methods of summability, Quart. J. Math. Oxford Ser., 72:244 256, 966. [3] S.M. Mazhar. N, p summability factors of ifiite series, Kodai Math. Sem. Rep., 8:96 00, 966. [4] B.E. Rhoades. Iclusio theorems for absolute matrix summability methods, J. Math. Aal. Appl., 238:82 90, 999. [5] N. Sigh ad N. Sharma. O N, p, q summability factors of ifiite series, Proc. Nat. Acad. Sci. Math. Sci., 0:6 68, 2000. [6] W.T. Sulaima. Relatios o some summability methods, Proc. Amer. Math. Soc., 8:39 45, 993.