ON THE QUASI MONOTONE AND ALMOST INCREASING SEQUENCES. 1. Introduction

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1 Journal of Mathematical Inequalities Volume 1, Number 4 (2007), ON THE QUASI MONOTONE AND ALMOST INCREASING SEQUENCES H. BOR AND H. S. ÖZARSLAN (communicated by G. Toader) Abstract. In this paper, a theorem of Bor and Özarslan [3] dealing with C, α; β k summability factors has been generalized for C, α, γ ; β k summability methods. 1. Introduction We will use the following notations and notions in our paper: If g > 0, then f = O(g) means that f < K.g, for some constant K > 0 (see [7]). Let(u n ) be a sequence. We write that Δu n = u n u n+1, Δ 0 u n = u n and Δ k u n = ΔΔ k 1 u n, for k = 1, 2,..., (see [7]). Abel s transformation ([8]): Let (a k ), (b k ) be complex sequences, and write S n = a 1 + a a n.then n 1 a k b k = S k Δb k + S n b n. (1) 1 Hölder s inequality ([8]): If p > 1, p + 1 q = 1anda 1, a 2, a 3,..., a n 0; b 1, b 2, b 3,..., b n 0,then ( ) 1/p ( 1/q a k b k bk) q. (2) a p k A sequence (b n ) of positive numbers is said to be δ -quasi-monotone, if b n 0, b n > 0 ultimately and Δb n δ n,where (δ n ) is a sequence of positive numbers (see [2]). A positive sequence (b n ) is said to be almost increasing if there exists a positive increasing sequence c n and two positive constants A and B such that Ac n b n Bc n (see [1]). Let a n be a given infinite series with partial sums (s n ). We denote by σn α and tn α the n-th Cesàro means of order α, with α > 1, of the sequence (s n) and (na n ), respectively, i.e., σ α n = 1 v=0 n v s v (3) Mathematics subject classification (2000): 40D15, 40F05, 40G05. Key words and phrases: Absolute summability, almost increasing and quasi-monotone sequences. c D l,zagreb Paper JMI

2 530 H. BOR AND H. S. ÖZARSLAN where t α n = 1 n v va v, (4) An α = O(nα ), α > 1, A0 α = 1 and Aα n = 0 for n> 0. (5) [5]) The series a n is said to be summable C, α k, k 1andα > 1, if (see n k 1 σn α 1 σα n 1 k = n tα n k < (6) and it is said to be summable C, α; β k, k 1, α > 1 andβ 0,if(see [6]) n βk+k 1 σn α σn 1 α k = n βk 1 tn α k <. (7) The series a n is said to be summable C, α, γ ; β k, k 1andα > 1, δ 0andγ is a real number, if (see [9]) n γ (βk+k 1) k tn α k <. (8) If we take γ = 1,then C, α, γ ; β k summability reduces to C, α; β k summability. Bor and Özarslan [3] have proved the following theorem for C, α; β k summability factors. THEOREM A. Let (X n ) be an almost increasing sequence such that ΔX n = O( Xn n ) and λ n 0 as n. Suppose that there exists a sequence of numbers (B n ) such that it is δ -quasi-monotone with nx n δ n <, Bn X n is convergent and Δλ n B n for all n. If the sequence (un α ),defined by (see [10]) satisfies the condition { t α un α n, α = 1 = max 1 v n tα v, 0 < α < 1 (9) n βk 1 (un α )k = O(X m ) as m, (10) then the series a n λ n is summable C, α; β k,k 1 and 0 β < α 1.

3 ON THE QUASI-MONOTONE AND ALMOST INCREASING SEQUENCES The main result The aim of this paper is to generalize Theorem A for C, α, γ ; β k summability factors. We shall prove the following theorem. THEOREM. Let (X n ) be an almost increasing sequence such that ΔX n = O( Xn n ) and λ n 0 as n. Suppose that there exists a sequence of numbers (B n ) such that it is δ -quasi-monotone with nx n δ n <, Bn X n is convergent and Δλ n B n for all n. If the sequence (un α ),defined by (9) satisfies the condition n γ (βk+k 1) k (un α ) k = O(X m ) as m, (11) then the series a n λ n is summable C, α, γ ; β k,where k 1, β 0, 0 < α 1 and γ is a real number such that k + αk γ (βk + k 1) > 1. We need the following lemmas for the proof of our theorem. LEMMA 1. ([4]) If 0 < α 1 and 1 v n,then v n p a p max m m pa p. (12) 1 m v p=0 p=0 LEMMA 2. ([3]) Under the conditions regarding (λ n ) and (X n ) of the Theorem, we have λ n X n as n. (13) LEMMA 3. ([3]) Under the conditions pertaining to (X n ) and (B n ) of the Theorem, we have that nb n X n (14) nx n ΔB n <. (15) 3. Proof of the Theorem Let (T α n ) be the n-th (C, α) mean of the sequence (na nλ n ). Then, by (4) we have T α n = 1 Using Abel s transformation, we get that T α n = 1 n 1 Δλ v v p=1 n v va vλ v. (16) n p pa p + λ n n v va v,

4 532 H. BOR AND H. S. ÖZARSLAN so that making use of Lemma 1, we have Since T α n 1 1 n 1 Δλ v v p=1 n p pa p + λ n An α n 1 Av α wv α Δλ v + λ n wn α = T α n,1 + Tα n,2, say. T α n,1 + Tα n,2 k 2 k ( T α n,1 k + T α n,2 k ), to complete the proof of the Theorem, it is sufficient to show that n γ (βk+k 1) k Tn,r α k < for r = 1, 2, by (8). n v va v Now, when k > 1, applying Hölder s inequality with indices k and k,where 1 k + 1 k = 1, we get that { } m+1 m+1 n 1 k n γ (βk+k 1) k Tn,1 α k n γ (βk+k 1) k 1 Av α uv α Δλ v n=2 n=2 n=2 { m+1 n 1 } n γ (βk+k 1) k αk v αk (uv α ) k B v { n 1 B v } k 1 v αk (uv α ) k B v v αk (uv α ) k B v m+1 n=v+1 v B v v γ (βk+k 1) k (uv α )k Δ(v B v ) +O(1)m B m v 1 nk+αk γ (βk+k 1) dx xαk+k γ (βk+k 1) v r γ (βk+k 1) k (ur α ) k r=1 v γ (βk+k 1) k (uv α )k Δ(v B v ) X v + O(1)m B m X m

5 ON THE QUASI-MONOTONE AND ALMOST INCREASING SEQUENCES 533 v ΔB v X v + O(1) B v+1 X v+1 +O(1)m B m X m as m, by virtue of the hypotheses of the Theorem and Lemma 3. Again, since λ n = O(1/X n )=O(1) by (13),wehavethat n γ (βk+k 1) k Tn,2 α k = λ n k 1 λ n n γ (βk+k 1) k (un α )k λ n n γ (βk+k 1) k (un α ) k Δ λ n +O(1) λ m v γ (βk+k 1) k (uv α ) k n γ (βk+k 1) k (un α )k Δλ n X n + O(1) λ m X m B n X n + O(1) λ m X m by virtue of the hypotheses of the Theorem, Lemma 2 and Lemma 3. Therefore, we get that n γ (βk+k 1) k Tn,r α k as m, for r = 1, 2. as m, This completes the proof of the Theorem. If we take γ = 1, then we get Theorem A. In this case condition (11) reduces to condition (10). Alsoifwetakeγ = 1andβ = 0, then we have a new result concerning C, α k summability factors. Finally if we take γ = 1, β = 0andα = 1, thenwe obtain a new result related to C, 1 k summability factors. Acknowledgement. The authors are very grateful to the referee for his/her invaluable suggestions.

6 534 H. BOR AND H. S. ÖZARSLAN REFERENCES [1] S.ALJANCIC AND D. ARANDELOVIC, O -regularly varying functions, Publ. Inst. Math., 22 (1977), [2] R.P. BOAS, Quasi-positive sequences and trigonometric series, Proc. London Math. Soc. Ser. A, 14 (1965), [3] H. BOR AND H.S. ÖZARSLAN, A note on absolute summability factors, Adv. Stud. Contemp. Math. (Kyungshang), 6(2003), [4] L.S. BOSANQUET, A mean value theorem, J. London Math. Soc., 16 (1941), [5] T.M. FLETT, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc., 7 (1957), [6] T.M. FLETT, Some more theorems concerning the absolute summability of Fourier series, Proc. London Math. Soc., 8 (1958), [7] G.H. HARDY, Divergent Series, Oxford Univ. Press, Oxford, (1949). [8] I.J.MADDOX, Introductory Mathematical Analysis, Adam Hilger Ltd., Bristol, (1977). [9] A.N. TUNCER, On generalized absolute Ces à ro summability factors, Ann. Polon. Math., 78 (2002), [10] T. PATI, The summability factors of infinite series, Duke Math. J., 21 (1954), (Received July 28, 2007) H. Bor Department of Mathematics Erciyes University Kayseri Turkey bor@erciyes.edu.tr H. S. Özarslan Department of Mathematics Erciyes University Kayseri Turkey seyhan@erciyes.edu.tr Journal of Mathematical Inequalities jmi@ele-math.com