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Almost increasing sequences and their new applications

Abstract

In this paper, we generalize a known theorem dealing with |C,1 | k summability factors to the | C , α | k summability factors of infinite series using an almost increasing sequence. This theorem also includes some known and new results.

MSC:26D15, 40D15, 40F05, 40G05.

1 Introduction

A positive sequence ( b n ) is said to be an almost increasing sequence if there exists a positive increasing sequence ( c n ) and two positive constants A and B such that A c n b n B c n (see [1]). Let a n be a given infinite series with the sequence of partial sums ( s n ). By t n α we denote the n th Cesàro mean of order α, with α>1, of the sequence (n a n ), that is,

t n α = 1 A n α v = 0 n A n v α 1 v a v ,
(1)

where

A n α = ( n + α n ) = ( α + 1 ) ( α + 2 ) ( α + n ) n ! =O ( n α ) , A n α =0for n>0.
(2)

The series a n is said to be summable |C,α | k , k1, if (see [2])

n = 1 1 n | t n α | k <.
(3)

If we take α=1, then | C , α | k summability reduces to |C,1 | k summability.

2 Known result

Many works dealing with an application of almost increasing sequences to the absolute Cesàro summability factors of infinite series have been done (see [311]). Among them, in [10], the following main theorem dealing with |C,1 | k summability factors has been proved.

Theorem A Let ( φ n ) be a positive sequence and ( X n ) be an almost increasing sequence. If the conditions

n = 1 n| Δ 2 λ n | X n <,
(4)
| λ n | X n =O(1) as n,
(5)
φ n =O(1) as n,
(6)
nΔ φ n =O(1) as n,
(7)
v = 1 n | t v | k v X v k 1 =O( X n ) as n
(8)

are satisfied, then the series a n λ n φ n is summable |C,1 | k , k1.

3 The main result

The aim of this paper is to generalize Theorem A to the | C , α | k summability in the following form.

Theorem Let ( φ n ) be a positive sequence and let ( X n ) be an almost increasing sequence.

If the conditions (4), (5), (6) and (7) are satisfied, and the sequence ( w n α ) defined by (see [12])

w n α ={ | t n α | , α = 1 , max 1 v n | t v α | , 0 < α < 1 ,
(9)

satisfies the condition

v = 1 n ( w v α ) k v X v k 1 =O( X n ) as n,
(10)

then the series a n λ n φ n is summable | C , α | k , 0<α1, (α1)k>1 and k1.

Remark It should be noted that if we take α=1, then we get Theorem A. In this case, condition (10) reduces to condition (8) and the condition ‘(α1)k>1’ is trivial.

We need the following lemmas for the proof of our theorem.

Lemma 1 [13]

If 0<α1 and 1vn, then

| p = 0 v A n p α 1 a p | max 1 m v | p = 0 m A m p α 1 a p | .
(11)

Lemma 2 [14]

Under the conditions (4) and (5), we have

n X n |Δ λ n |=O(1) as n,
(12)
n = 1 X n |Δ λ n |<.
(13)

4 Proof of the Theorem

Let ( T n α ) be the n th (C,α) mean, with 0<α1, of the sequence (n a n λ n φ n ).

Then, by (1), we find that

T n α = 1 A n α v = 1 n A n v α 1 v a v λ v φ n .
(14)

Thus, applying Abel’s transformation first and then using Lemma 1, we have that

T n α = 1 A n α v = 1 n 1 Δ ( λ v φ n ) p = 1 v A n p α 1 p a p + λ n φ n A n α v = 1 n A n v α 1 v a v = 1 A n α v = 1 n 1 ( λ v Δ φ v + φ v + 1 Δ λ v ) p = 1 v A n p α 1 p a p + λ n φ n A n α v = 1 n A n v α 1 v a v , | T n α | 1 A n α v = 1 n 1 | λ v Δ φ v | | p = 1 v A n p α 1 p a p | + 1 A n α v = 1 n 1 | φ v + 1 Δ λ v | | p = 1 v A n p α 1 p a p | + | λ n φ n | A n α | v = 1 v A n v α 1 v a v | 1 A n α v = 1 n 1 A v α w v α | λ v | | Δ φ v | + 1 A n α v = 1 n 1 A v α w v α | φ v + 1 | | Δ λ v | + | λ n | | φ n | w n α = T n , 1 α + T n , 2 α + T n , 3 α .

To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that

n = 1 n 1 | T n , r α | k <for r=1,2,3.

Now, when k>1, applying Hölder’s inequality with indices k and k , where 1 k + 1 k =1, we get that

n = 2 m + 1 n 1 | T n , 1 α | k n = 2 m + 1 n 1 ( A n α ) k { v = 1 n 1 A v α w v α | Δ φ v | | λ v | } k = O ( 1 ) n = 2 m + 1 1 n 1 + α k v = 1 n 1 ( v α ) k ( w v α ) k | Δ φ v | k | λ v | k { v = 1 n 1 1 } k 1 = O ( 1 ) n = 2 m + 1 1 n 2 + ( α 1 ) k v = 1 n 1 v α k ( w v α ) k | λ v | k 1 v k = O ( 1 ) v = 1 m v α k ( w v α ) k v k | λ v | k n = v + 1 m + 1 1 n 2 + ( α 1 ) k = O ( 1 ) v = 1 m v α k ( w v α ) k v k | λ v | k v d x x 2 + ( α 1 ) k = O ( 1 ) v = 1 m ( w v α ) k | λ v | | λ v | k 1 1 v = O ( 1 ) v = 1 m ( w v α ) k | λ v | 1 v X v k 1 = O ( 1 ) v = 1 m 1 Δ | λ v | r = 1 v ( w r α ) k r X r k 1 + O ( 1 ) | λ m | v = 1 m ( w v α ) k v X v k 1 = O ( 1 ) v = 1 m | Δ λ v | X v + O ( 1 ) | λ m | X m = O ( 1 ) as  m

by virtue of the hypotheses of the theorem and Lemma 2. Again, we get that

n = 2 m + 1 n 1 | T n , 2 α | k n = 2 m + 1 n 1 ( A n α ) k { v = 1 n 1 A v α w v α | φ v + 1 | | Δ λ v | } k = O ( 1 ) n = 2 m + 1 1 n 1 + α k { v = 1 n v α ( w v α ) | Δ λ v | } k = O ( 1 ) n = 2 m + 1 1 n 1 + α k v = 1 n 1 v α k ( w v α ) k | Δ λ v | k { v = 1 n 1 1 } k 1 = O ( 1 ) n = 2 m + 1 1 n 2 + ( α 1 ) k v = 1 n 1 v α k ( w v α ) k | Δ λ v | k = O ( 1 ) v = 1 m v α k ( w v α ) k | Δ λ v | | Δ λ v | k 1 n = v + 1 m + 1 1 n 2 + ( α 1 ) k = O ( 1 ) v = 1 m v α k ( w v α ) k | Δ λ v | v k 1 X v k 1 v d x x 2 + ( α 1 ) k = O ( 1 ) v = 1 m v | Δ λ v | ( w v α ) k v X v k 1 = O ( 1 ) v = 1 m Δ ( v | Δ λ v | ) r = 1 v ( w r α ) k r X r k 1 + O ( 1 ) m | Δ λ m | v = 1 m ( w v α ) k v X v k 1 = O ( 1 ) v = 1 m 1 v | Δ 2 λ v | X v + O ( 1 ) v = 1 m 1 X v | Δ λ v | + O ( 1 ) m | Δ λ m | X m = O ( 1 ) as  m

by hypotheses of the theorem and Lemma 2. Finally, as in T n , 1 α , we have that

n = 1 m n 1 | T n , 3 α | k = n = 1 m n 1 | λ n φ n w n α | k = O ( 1 ) n = 1 m ( w n α ) k | λ n | n X n k 1 = O ( 1 ) as  m

by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. Also, if we take k=1, then we get a new result concerning the |C,α| summability factors of infinite series.

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

The author expresses his thanks to the referees for their useful comments and suggestions.

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Bor, H. Almost increasing sequences and their new applications. J Inequal Appl 2013, 207 (2013). https://doi.org/10.1186/1029-242X-2013-207

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Keywords

  • increasing sequences
  • Cesàro mean
  • summability factors
  • Hölder inequality
  • Minkowski inequality