International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in TMIMT, Moradabad, U. P., India ssumath@yahoo.co.in Abstract In the present paper, a general theorem concerning ϕ C, α, ρ, γ k summability factors of infinte series, has been proved. Mathematics Subject Classification: 40D15, 40F05, 40G05 Keywords: Absolute Summability, Almost Increasing Sequence, Quasi- Monotone Sequence 1 Introduction A sequence of (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[3]). A positive sequence (b n ) is said to be almost increasing if there exists a psitive increasing sequence (c n ) and two positive constants A and B such that Ac n b n Bc n (see[1]). Let (ϕ n ) be a sequence of complex numbers and let a n be a given infinite series with partial sums (s n ). We denote by σn α and tα n 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 A α n ν=0 n ν s ν (1) where t α n = 1 A α n n ν νa ν, (2) A α n = O (n α ), α > 1, A α 0 =1and A α n =0for n > 0. (3)
144 S. K. Saxena The series a n is said to be summable C, α k, k 1 and α> 1 if (see[7]) n k 1 σ α n σn 1 α k <. (4) But since t α n = n ( σ α n σ α n 1) (see[9]) condition (4) can also be written as 1 n tα n k <. (5) The series a n is said to be summable C, α, ρ, γ k, k 1, if (see[8]) n γ(ρk+k 1) k t α n k <, (6) where ρ 0 and γ is a real number. The series a n is said to be summable ϕ C, α k, k 1 and α> 1, if (see [2] and [10]) n k ϕ n t n k <, (7) and it is said to be summable ϕ C, α, ρ, γ k, k 1, ρ 0, γ 1if n γ(ρk+k 1) 2k+1 ϕ n t α n k <. (8) In the special case when γ = 1 and ρ =0,ϕ C, α, ρ, γ k summability is same as ϕ C, α k summability and when ϕ n = n 1 1 k (resp. ϕ n = n β+1 1 k ) ϕ C, α, ρ, γ k summability is same as C, α, ρ, γ k summability. 2 Known Result Mazhar [12] proved the following theorem for C, 1 k summability factors of infinite series. Theorem 2.1 Let λ n 0 as n. Suppose that there exists a sequence of numbers (B n ) such that it is δ-quasi monotone with nδ n log n<, Bn log n is convergent and Δλ n B n for all n. If 1 n t n k = O (log m) as m, (9) Later on Bor and Leindler [4] generalized the above theorem under weaker conditions in the following form for ϕ C, α k summability
δ-quasi monotone and increasing sequence 145 Theorem 2.2 Let (X n ) be an almost increasing sequence such that ΔX n = O ( ) X n n and λn 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 <, B n X n is convergent and Δλ n B n for all n. If there exists an ε>0 such that the sequence ( n ε k ϕ n k) is non-increasing and if the sequence (ωn), α defined by (see[13]) ωn α = t α n, α =1 max 1 ν n tα ν, 0 <α<1 (10) satisfies the condition n k (ωn α ϕ n ) k = O (X m ) as m, (11) then the series a n λ n is summable ϕ C, α k, k 1, 1 k α 1. 3 Main Result The aim of this paper is to generalize Theorem 2.2 for ϕ C, α, ρ, γ k summability in the following form. Theorem 3.1 Let (X n ) be an almost increasing sequence such that ΔX n = O ( ) X n n as λn 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 <, B n X n is convergent ( and Δλ n B n for all n. If there exists an ε>0 such that the sequence n ε k ϕ n k) is non-increasing and if the sequence (ωn α ) defined by (see [13]) as (10) satisfies the condition n γ(ρk+k 1) 2k+1 ( ϕ n ωn) α k = O (X m ) as m, (12) then the series a n λ n is summable ϕ C, α, ρ, γ k,k 1, 1 α 1. k It is also a generalization of Bor and Özarslan [5] We need the following lemmas for the proof of our theorem. Lemma 3.2 [6] If 0 <α 1 and 1 ν n, then n pa p max m pa p. (13) 1 m ν p=0 Lemma 3.3 Under the conditions regarding (λ n ) and (X n ) of the Theorem, we have λ n X n = O (1) as n, (14) p=0
146 S. K. Saxena The statements proof of Lemma 3.3 are proved by Bor and Leindler [4] and hence is omitted. Lemma 3.4 Under the conditions pertaining to (X n ) and (B n ) of the Theorem, we have nb n X n = O (1) (15) nx n ΔB n <. (16) The statements proof of Lemma 3.4 are proved in Theorem 1 and Theorem 2 of Leindler [11] and hence is omitted. Proof of Theorem 3.1 Let (T α n ) be the n-th (C, α), mean of the sequence (na n λ n ). Then, by (2), we have Tn α = 1 A α n Using Abel s transformation, we get Tn α = 1 n 1 Δλ A α ν n p=1 n ννa ν λ ν. n p pa p + λ n A α n n ν νa ν, so that making use of Lemma 1, we have Tn α 1 n 1 Δλ A α ν n p pa p + λ n n p=1 A α n ν νa ν n 1 n 1 A α A α ν ων α Δλ ν + λ n ωn α n = Tn,1 α + T n,2 α, say. Since Tn,1 α + T n, α k ( T 2 k α k T n,1 + α k ) n,1, to complete the proof of the theorem, it is sufficient to show that n γ(ρk+k 1) 2k+1 ϕn Tn,r α k <, for r =1, 2 by (8). Now, when k>1, applying Hölder s inequality with indices k and k, where = 1, we get that 1 k + 1 k n γ(ρk+k 1) 2k+1 ϕn Tn,1 α k n=2 { n 1 n γ(ρk+k 1) 2k+1 (A α n ) k ϕ n k A α ν ωα ν Δλ ν } k
δ-quasi monotone and increasing sequence 147 { n 1 } = O (1) n γ(ρk+k 1) 2k+1 n αk ϕ n k ν αk (ων α )k B ν n=2 { n 1 } k 1 B ν = O (1) ν αk (ω α ϕ n k ν )k B ν nγ(ρk+k 1) 2k+1 n=ν+1 nαk m+1 = O (1) ν αk (ων α n ε+γ(ρk+k 1) 2k+1 )k B ν ϕ n=ν+1 n αk+ε n k = O (1) ν αk (ων α ) k B ν ν ε+γ(ρk+k 1) 2k+1 ϕ ν k 1 n=ν+1 n αk+ε = O (1) ν αk (ων α )k B ν ν ε+γ(ρk+k 1) 2k+1 ϕ ν k dx ν x αk+ε = O (1) ν B ν ν γ(ρk+k 1) 2k+1 (ων α ϕ ν ) k = O (1) Δ(ν B ν ) r γ(ρk+k 1) 2k+1 (ωr α ϕ r ) k r=1 + O (1) m B m ν γ(ρk+k 1) 2k+1 (ων α ϕ ν ) k = O (1) (ν +1) ΔB ν B ν X ν + O (1) m B m X m = O (1) ν ΔB ν X ν + O (1) B ν X ν + O (1) m B m X m = O (1) ν ΔB ν X ν + O (1) B ν+1 X ν+1 + O (1) m B m X m = O (1) as m by the virtue of the hypotheses of the Theorem 3.1 and Lemma 3.4. Again, since λ n = O ( 1 X n ) = O (1), by (14) we have n γ(ρk+k 1) 2k+1 ϕn Tn,2 α k = λ n k 1 λ n n γ(ρk+k 1) 2k+1 (ωn α ϕ n ) k = O (1) λ n n γ(ρk+k 1) 2k+1 (ωn α ϕ n ) k
148 S. K. Saxena = O (1) Δ λ n ν γ(ρk+k 1) 2k+1 (ων α ϕ ν ) k + O (1) λ m n γ(ρk+k 1) 2k+1 (ωn α ϕ n ) k = O (1) Δ λ n X n + O (1) λ m X m = O (1) B n X n + O (1) λ m X m = O (1) as m, by the virtue of the hypotheses of Theorem 3.1 and Lemma 3.3. Therefore, we get that n γ(ρk+k 1) 2k+1 ϕn Tn,r α k = O (1), as m for r =1, 2. This completes the proof of the Theorem 3.1. References [1] S. Aljančić and D. Arandelovic, O-regular varying functions, Publ. Inst. Math., 22 (1977), 5-22. [2] M. Balci, Absolute ϕ-summability factors, Comm. Fac. Sci. Univ. Ankara, Ser. A, 29 (1980), 63-80. [3] R. P. Boas, Quasi positive sequences and trigonometric series, Proc. London Math. Soc. Ser. A, 14 (1965), 38-46. [4] H. Bor and L. Leindler, A note on δ-quasi-monotone and almost increasing sequences, Mathematical inequalities and Applications, 8(1) (2005), 129-134. [5] H. Bor and H. S. Özarslan, On the quasi-monotone and almost increasing sequences, Journal of Mathematical Inequalities, 1(4) (2007), 529-534. [6] L. S. Bosanquit, A mean value theorem, J. London Math. Soc., 16 (1941), 146-148. [7] T. M. Flett, On an extension of absolute summability and some theorems of Littelwood and Paley, Proc. London Math. Soc., 7 (1957), 113-141. [8] A. N. Gürkan, On absolute Cesáro summability factors, J. Anal., 7 (1999), 133-138.
δ-quasi monotone and increasing sequence 149 [9] E. Kogbetliantz, Sur la series absolument summability methode des moyennes arithmetiques, Bull. Sci. Math., 49 (1925), 234-256. [10] L. Leindler, On extension of some theorems of Flett I, Acta. Math. Hungar, 64(3) (1994), 2 9-283. [11] L. Leindler, Three theorems connected with δ-quasi monotone sequences and their applications to an integrability theorem, Publ. Math. (Debrecen), 60 (2002), 373-384. [12] S. M. Mazhar, On a generalized quasi-convex sequence and its applications, Indian J. pure appl. Math. 8 (1977), 784-790. [13] T. Pati, The summability factors of infinite series, Duke Math. J., 21 (1954), 271-284. Received: October 11, 2008