Journal of Inequalities in Pure and Applied Mathematics A NOTE ON SUMMABILITY FACTORS S.M. MAZHAR Department of Mathematics and Computer Science Kuwait University P.O. Box No. 5969, Kuwait - 13060. volume 7, issue 4, article 135, 2006. Received 18 March, 2006; accepted 03 July, 2006. Communicated by: H. Bor EMail: sm_mazhar@hotmail.com Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 081-06 Quit Abstract In this note we investigate the relation between two theorems proved by Bor [2, 3] on |N̄, pn |k summability of an infinite series. 2000 Mathematics Subject Classification: 40D15, 40F05, 40G05. Key words: Absolute summability factors. A Note on Summability Factors Contents S.M. Mazhar 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 3 5 7 Title Page Contents JJ J II I Go Back Close Quit Page 2 of 11 J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au 1. Introduction P Let an be a given infinite series with {sn } as the sequence of its n-th partial sums. Let {pn } be a sequence of positive constants such that Pn = p0 + p1 + p2 + · · · + pn −→ ∞ as n −→ ∞. Let n 1 X tn = p ν sν . Pn ν=1 P P The series an is said to be summable |N̄ , pn | if ∞ 1 |tn − tn−1 | < ∞. It is said to be summable |N̄ , pn |k , k ≥ 1 [1] if k−1 ∞ X Pn (1.1) 1 pn |tn − tn−1 |k < ∞, n X pν |sν |k = O(Pn ), n −→ ∞. Title Page Concerning |N̄ , pn | summability factors of lowing theorem: P an , T. Singh [6] proved the fol- Theorem A. If the sequences {pn } and {λn } satisfy the conditions ∞ X 1 JJ J II I Go Back 1 (1.3) S.M. Mazhar Contents and bounded [N̄ , pn ]k , k ≥ 1 if (1.2) A Note on Summability Factors Close Quit Page 3 of 11 J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 pn |λn | < ∞, http://jipam.vu.edu.au Pn |∆λn | ≤ Cpn |λn |, (1.4) C is a constant, and if |N̄ , pn |. P an is bounded [N̄ , pn ]1 , then P an Pn λn is summable Earlier in 1968 N. Singh [5] had obtained the following theorem. P Theorem B. If an is bounded [N̄ , pn ]1 and {λn } is a sequence satisfying the following conditions ∞ X pn |λn | (1.5) 1 Pn A Note on Summability Factors S.M. Mazhar < ∞, Title Page Contents Pn ∆λn = O(|λn |), pn (1.6) then P an λn is summable |N̄ , pn |. JJ J II I Go Back In order to extend these theorems to the summability |N̄ , pn |k , k ≥ 1, Bor [2, 3] proved the following theorems. P Theorem C. Under the conditions (1.2), (1.3) and (1.4), the series an Pn λn is summable |N̄ , pn |k , k ≥ 1. P Theorem D. If an is bounded [N̄ , pn ]k , kP≥ 1 and {λn }, is a sequence satisfying the conditions (1.4) and (1.5), then an λn is summable |N̄ , pn |k . Close Quit Page 4 of 11 J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au 2. Results In this note we propose to examine the relation between Theorem C and Theorem D. We recall that recently Sarigol and Ozturk [4] constructed an example to demonstrate that P the hypotheses of Theorem A are not sufficient for the summability |N̄ , pn | of an Pn λn . They proved that Theorem A holds true if we assume the additional condition (2.1) pn+1 = O(pn ). A Note on Summability Factors S.M. Mazhar From (1.4) we find that ∆λn Cpn λ n+1 λn = 1 − λn ≤ Pn , Hence λn+1 λn+1 = − 1 + 1 λn λn λn+1 ≤ 1 − +1 λn Cpn ≤ + 1 ≤ C. Pn Thus |λn+1 | ≤ C|λn |, and combining this with (2.1) we get Title Page Contents JJ J II I Go Back Close Quit Page 5 of 11 J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au (2.2) pn+1 |λn+1 | ≤ Cpn |λn |. Clearly (2.1) and (1.4) imply (2.2). However (2.2) need not imply (2.1) or (1.4). In view of ∆(Pn λn ) = Pn ∆λn − pn+1 λn+1 it is clear that if (2.2) holds, then the condition (1.4) is equivalent to the condition |∆(Pn λn )| ≤ Cpn |λn |. (2.3) It can be easily verified that a corrected version of Theorem A and Theorem C and also a slight generalization of the result of Sarigol and Ozturk for k = 1 can be stated as P Theorem 2.1. Under the conditions (1.2), (1.3) (2.2) and (2.3) the series an Pn λn is summable |N̄ , pn |k , k ≥ 1 We now proceed to show that Theorem 2.1 holds good without condition (2.2). Thus we have: P Theorem 2.2. Under the conditions (1.2), (1.3) and (2.3) the series an Pn λn is summable |N̄ , pn |k , k ≥ 1. A Note on Summability Factors S.M. Mazhar Title Page Contents JJ J II I Go Back Close To prove Theorem 2.2 we first prove the following lemma. Lemma 2.3. Under the conditions of Theorem 2.2 (2.4) m X 1 pn |λn ||sn |k = O(1) as m −→ ∞. Quit Page 6 of 11 J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au 3. Proofs Proof of Lemma 2.3. In view of (1.3) and (2.3) ∞ X |∆(λn Pn )| ≤ C 1 ∞ X pn |λn | < ∞, 1 so it follows that {Pn λn } ∈ BV and hence Pn |λn | = O(1). Now m X 1 pn λn ||sn |k = m−1 X ∆|λn | 1 = O(1) pν |sν |k + |λm | ν=1 m−1 X m X pν |sν |k S.M. Mazhar ν=1 Title Page |∆λn |Pn + O(|λm |Pm ) 1 m−1 X = O(1) = O(1) n X A Note on Summability Factors Contents ! |∆(Pn λn )| + pn+1 |λn+1 | 1 m−1 X m X 1 1 pn |λn | + O(1) + O(1) pn+1 |λn+1 | + O(1) = O(1). JJ J II I Go Back Close Quit Page 7 of 11 P Proof of Theorem 2.2. Let Tn denote the nth N̄ , pn means of the series an Pn λn . J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au Then n ν 1 X X Tn = pν ar P r λ r Pn ν=0 r=0 n 1 X = (Pn − Pν−1 ) aν Pν λν . Pn ν=0 so that for n ≥ 1 A Note on Summability Factors Tn − Tn−1 = = pn Pn Pn−1 pn Pn Pn−1 n X ν=1 n−1 X Pν−1 aν Pν λν ∆(Pν−1 Pν λν )sν + pn λn sn ν=1 = L1 + L2 , say. Thus to prove the theorem it is sufficient to show that k−1 ∞ X Pn 1 pn S.M. Mazhar Title Page Contents JJ J II I Go Back k |Lν | < ∞, ν = 1, 2. Close Quit Now Page 8 of 11 |∆(Pν−1 Pν λν )| ≤ pν Pν |λν | + Pν |∆(Pν λν )| ≤ Cpν Pν |λν | J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au in view of (2.3). So m+1 X n=2 Pn pn k−1 k |L1 | = O(1) m+1 X n=2 = O(1) = O(1) m+1 X n=2 m+1 X n=2 = O(1) m X n−1 X pn k Pn Pn−1 pn k Pn Pn−1 pn Pn Pn−1 !k pν Pν |λν ||sν | ν=1 n−1 X ! (Pν |λν |)k |sν |k pν ν=1 n−1 X n−1 X !k−1 pν ν=1 Pν |λν |pν |sν |k ν=1 A Note on Summability Factors S.M. Mazhar pν |λν |sν |k = O(1) Title Page ν=1 in view of the lemma and Pn |λn | = O(1). Also m+1 m+1 X Pn k−1 X |L2 |k = O(1) pn |λn |k |sn |k Pnk−1 pn 1 1 = O(1) m+1 X pn |λn ||sn |k 1 = O(1). This proves Theorem 2.2. Contents JJ J II I Go Back Close Quit Page 9 of 11 J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au Thus a generalization of a corrected version of Theorem C is Theorem 2.2. Writing λn = µn Pn the conditions (1.5) and (1.4) become (3.1) ∞ X pn |µn | < ∞, 1 (3.2) |∆(Pn µn )| ≤ Cpn |µn |, consequently Theorem D can be stated as: P If an is bounded [N̄ , Pn ]k , k ≥ 1 and {µn } is a sequence satisfying (3.1) P and (3.2) then an Pn µn is summable |N̄ , pn |k , k ≥ 1. Thus Theorem D is the same as Theorem 2.2 which is a generalization of the corrected version of Theorem C. A Note on Summability Factors S.M. Mazhar Title Page Contents JJ J II I Go Back Close Quit Page 10 of 11 J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au References [1] H. BOR, On |N̄ , pn |k summability methods and |N̄ , pn |k summability factors of infinite series, Ph.D. Thesis (1982), Univ. of Ankara. [2] H. BOR, On |N̄ , pn |k summability factors, Proc. Amer. Math. Soc., 94 (1985), 419–422. [3] H. BOR, On |N̄ , pn |k summability factors of infinite series, Tamkang J. Math., 16 (1985), 13–20. [4] M. ALI SARIGOL AND E. OZTURK, A note on |N̄ , pn |k summability factors of infinite series, Indian J. Math., 34 (1992), 167–171. [5] N. SINGH, On |N̄ , pn | summability factors of infinite series, Indian J. Math., 10 (1968), 19–24. [6] T. SINGH, A note on |N̄ , pn | summability factors of infinite series, J. Math. Sci., 12-13 (1977-78), 25–28. A Note on Summability Factors S.M. Mazhar Title Page Contents JJ J II I Go Back Close Quit Page 11 of 11 J. Ineq. Pure and Appl. Math. 7(4) Art. 135, 2006 http://jipam.vu.edu.au