A General Note on δquasi Monotone and Increasing Sequence


 Ανδρόνικος Λιάπης
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1 International Mathematical Forum, 4, 2009, no. 3, A General Note on δquasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in TMIMT, Moradabad, U. P., India 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 δquasi 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 nth 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)
2 144 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 δquasi 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
3 δquasi 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 δquasi 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 nonincreasing 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 δquasi 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 nonincreasing 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
4 146 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 nth (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
5 δquasi 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
6 148 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, Oregular varying functions, Publ. Inst. Math., 22 (1977), [2] M. Balci, Absolute ϕsummability factors, Comm. Fac. Sci. Univ. Ankara, Ser. A, 29 (1980), [3] R. P. Boas, Quasi positive sequences and trigonometric series, Proc. London Math. Soc. Ser. A, 14 (1965), [4] H. Bor and L. Leindler, A note on δquasimonotone and almost increasing sequences, Mathematical inequalities and Applications, 8(1) (2005), [5] H. Bor and H. S. Özarslan, On the quasimonotone and almost increasing sequences, Journal of Mathematical Inequalities, 1(4) (2007), [6] L. S. Bosanquit, A mean value theorem, J. London Math. Soc., 16 (1941), [7] T. M. Flett, On an extension of absolute summability and some theorems of Littelwood and Paley, Proc. London Math. Soc., 7 (1957), [8] A. N. Gürkan, On absolute Cesáro summability factors, J. Anal., 7 (1999),
7 δquasi monotone and increasing sequence 149 [9] E. Kogbetliantz, Sur la series absolument summability methode des moyennes arithmetiques, Bull. Sci. Math., 49 (1925), [10] L. Leindler, On extension of some theorems of Flett I, Acta. Math. Hungar, 64(3) (1994), [11] L. Leindler, Three theorems connected with δquasi monotone sequences and their applications to an integrability theorem, Publ. Math. (Debrecen), 60 (2002), [12] S. M. Mazhar, On a generalized quasiconvex sequence and its applications, Indian J. pure appl. Math. 8 (1977), [13] T. Pati, The summability factors of infinite series, Duke Math. J., 21 (1954), Received: October 11, 2008
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