A new application of almost increasing sequences to factored infinite series
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A new application of almost increasing sequences to factored infinite series Hüseyin Bor1
· Ravi P. Agarwal2
Received: 20 March 2020 / Accepted: 12 May 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we proved a general theorem dealing with the absolute Cesàro summability factors by using an almost increasing sequence. This new theorem also contains as particular cases several known and new results on the absolute Cesàro summability factors of infinite series. Keywords Cesàro summability · Almost increasing sequence · Summability factors · Infinite series · Hölder’s inequality · Minkowski’s inequality Mathematics Subject Classification 26D15 · 40D15 · 40F05 · 40G05
1 Introduction A positive sequence (bn ) is said to be almost increasing sequence if there exists a positive increasing sequence (cn ) and two positive constants M and N such that Mcn ≤ bn ≤ N cn (see [2]). Forany sequence (λn ) we write that 2 λn = λn − λn+1 and λn = λn − λn+1 . Let an be a given infinite series. By tnα we denote the nth Cesàro mean of order α, with α > −1, of the sequence (nan ), that is (see [15])
tnα =
n 1 α−1 An−v vav , Aαn v=1
B
Hüseyin Bor [email protected] Ravi P. Agarwal [email protected]
1
Bahcelievler, Ankara, Turkey
2
Department of Mathematics, Texas A&M University-Kingsvilie, Kingsvilie, TX 78363, USA 0123456789().: V,-vol
(1)
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H. Bor, R. P. Agarwal
where Aαn =
(α + 1)(α + 2)...(α + n) = O(n α ), n!
Aα−n = 0 for n > 0.
(2)
Let (ϕn ) be a sequence of complex numbers and let (wnα ) be a sequence defined by (see [20]) α t , α=1 n (3) wnα = max1≤v≤n tvα , 0 < α < 1. The series
an is said to be summable ϕ − |C, α|k , k ≥ 1, if (see [1]) ∞
n −k | ϕn tnα |k < ∞.
(4)
n=1 1
In the special case when ϕn = n 1− k , ϕ− | C, α |k summability is the same as 1 | C, α |k summability (see [16]). If we take ϕn = n δ+1− k , then we obtain | C, α; δ |k summability (see [17]).
2 The known result As an application of almost increasing sequences, many works dealing with the absolute summability factors of infinite series have been done (see [3–13], [18,19], [21]). Among them, in [9], the following theorem has been proved dealing with the | C, α |k summability factors of infinite series. Theorem A Let (wnα ) be a sequence defined as in (3). Let (σn ) be a positive sequence and let (X n ) be an almost increasing sequence. If the conditions ∞ n 2 λn X n < ∞,
(5)
n=1
|λn | X n = O(1) as n → ∞,
(6)
σn = O(1) as n → ∞,
(7)
nσn = O(1) as n → ∞,
(8)
n (wvα )k v=1
hold, then the series
v X vk−1
= O(X n ) as n → ∞
an λn σn is summable | C, α |k , 0 < α ≤ 1, k ≥ 1.
(9)
A new application of almost increasing sequences to…
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3 The main result The aim of this paper is to generalize Theorem A to the ϕ− | C, α |k summability in the following form. Theorem Let (ϕn ) be a sequence of complex numbers and let (wnα ) be a sequence defined as in (3). Let (σn ) be a positive sequence and let (X n ) be an almost increasing sequenc
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