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On a new application of almost increasing sequences

Journal of Inequalities and Applications20132013:13

https://doi.org/10.1186/1029-242X-2013-13

  • Received: 3 September 2012
  • Accepted: 6 November 2012
  • Published:

Abstract

In (Bor in Int. J. Math. Math. Sci. 17:479-482, 1994), Bor has proved the main theorem dealing with | N ¯ , p n | k summability factors of an infinite series. In the present paper, we have generalized this theorem on the φ | A , p n | k summability factors under weaker conditions by using an almost increasing sequence instead of a positive non-decreasing sequence.

MSC:40D15, 40F05, 40G99.

Keywords

  • absolute matrix summability
  • almost increasing sequences
  • infinite series

1 Introduction

Let a n be a given infinite series with the partial sums ( s n ) . We denote by t n the n th ( C , 1 ) mean of the sequence ( s n ) . The series a n is said to be summable | C , 1 | k , k 1 , if (see [1])
n = 1 n k 1 | t n t n 1 | k < .
(1)
Let ( p n ) be a sequence of positive numbers such that
P n = v = 0 n p v as  n ( P i = p i = 0 , i 1 ) .
(2)
The sequence-to-sequence transformation
σ n = 1 P n v = 0 n p v s v
(3)
defines the sequence ( σ n ) of the ( N ¯ , p n ) mean of the sequence ( s n ) , generated by the sequence of coefficients ( p n ) (see [2]). The series a n is said to be summable | N ¯ , p n | k , k 1 , if (see [3])
n = 1 ( P n p n ) k 1 | σ n σ n 1 | k < .
(4)
Let A = ( a n v ) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence s = ( s n ) to A s = ( A n ( s ) ) , where
A n ( s ) = v = 0 n a n v s v , n = 0 , 1 , .
(5)
The series a n is said to be summable | A , p n | k , k 1 , if (see [4])
n = 1 ( P n p n ) k 1 | Δ ¯ A n ( s ) | k < ,
(6)
where
Δ ¯ A n ( s ) = A n ( s ) A n 1 ( s ) .
Let ( φ n ) be any sequence of positive real numbers. The series a n is said to be summable φ | A , p n | k , k 1 , if (see [5])
n = 1 φ n k 1 | Δ ¯ A n ( s ) | k < .
(7)

If we take φ n = P n p n , then φ | A , p n | k summability reduces to | A , p n | k summability. Also, if we take φ n = P n p n and a n v = p v P n , then we get | N ¯ , p n | k summability. Furthermore, if we take φ n = n , a n v = p v P n and p n = 1 for all values of n, φ | A , p n | k reduces to | C , 1 | k summability. Finally, if we take φ n = n and a n v = p v P n , then we get | R , p n | k summability (see [6]).

Before stating the main theorem, we must first introduce some further notations.

Given a normal matrix A = ( a n v ) , we associate two lower semimatrices A ¯ = ( a ¯ n v ) and A ˆ = ( a ˆ n v ) as follows:
a ¯ n v = i = v n a n i , n , v = 0 , 1 ,
(8)
and
a ˆ 00 = a ¯ 00 = a 00 , a ˆ n v = a ¯ n v a ¯ n 1 , v , n = 1 , 2 , .
(9)
It may be noted that A ¯ and A ˆ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then we have
A n ( s ) = v = 0 n a n v s v = v = 0 n a ¯ n v a v
(10)
and
Δ ¯ A n ( s ) = v = 0 n a ˆ n v a v .
(11)

2 Known result

Many works have been done dealing with | N ¯ , p n | k summability factors of infinite series (see [722]). Among them, in [21], the following main theorem has been proved.

Theorem A Let ( X n ) be a positive non-decreasing sequence and let there be sequences ( β n ) and ( λ n ) such that
(12)
(13)
(14)
(15)
are satisfied. Furthermore, if ( p n ) is a sequence of positive numbers such that
(16)
(17)

then the series a n λ n is summable | N ¯ , p n | k , k 1 .

3 The main result

The aim of this paper is to generalize Theorem A for φ | A , p n | k summability under weaker conditions. For this, we need the concept of an almost increasing sequence. A positive sequence ( c n ) is said to be almost increasing if there exists a positive increasing sequence ( b n ) and two positive constants A and B such that A b n c n B b n (see [23]). Obviously, every increasing sequence is an almost increasing sequence but the converse need not be true as can be seen from the example b n = n e ( 1 ) n . Also, one can find some results dealing with absolute almost convergent sequences (see [24]). So, we are weakening the hypotheses of Theorem A replacing the increasing sequence by an almost increasing sequence. Now, we shall prove the following theorem.

Theorem Let A = ( a n v ) be a positive normal matrix such that
(18)
(19)
(20)
(21)
Let ( X n ) be an almost increasing sequence and ( φ n p n P n ) be a non-increasing sequence. If conditions (12)-(16) and
n = 1 m φ n k 1 ( p n P n ) k | s n | k = O ( X m ) as m ,
(22)

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

Remark It should be noted that if we take ( X n ) as a positive non-decreasing sequence, φ n = P n p n and a n v = p v P n , then we get Theorem A. In this case, conditions (21) and (22) reduce to conditions (16) and (17), respectively. Also, the condition ‘ ( φ n p n P n ) is a non-increasing sequence’ and the conditions (18)-(20) are automatically satisfied.

Lemma [22]

Under the conditions on ( X n ) , ( β n ) and ( λ n ) as taken in the statement of the theorem, we have the following:
(23)
(24)
Proof of the Theorem Let ( T n ) denote A-transform of the series a n λ n . Then we have, by (10) and (11),
Δ ¯ T n = v = 1 n a ˆ n v λ v a v .
Applying Abel’s transformation to this sum, we get that
Δ ¯ T n = v = 1 n 1 Δ v ( a ˆ n v λ v ) s v + a ˆ n n λ n s n = v = 1 n 1 ( a ˆ n v λ v a ˆ n , v + 1 λ v + 1 ) s v + a ˆ n n λ n s n = v = 1 n 1 Δ v ( a ˆ n v ) λ v s v + v = 1 n 1 a ˆ n , v + 1 s v Δ λ v + a n n λ n s n = T n ( 1 ) + T n ( 2 ) + T n ( 3 ) .
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
n = 1 φ n k 1 | T n ( r ) | 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 have that
n = 2 m + 1 φ n k 1 | T n ( 1 ) | k = O ( 1 ) n = 2 m + 1 φ n k 1 ( v = 1 n 1 | Δ v ( a ˆ n v ) | | λ v | | s v | ) k = O ( 1 ) n = 2 m + 1 φ n k 1 ( v = 1 n 1 | Δ v ( a ˆ n v ) | | λ v | k | s v | k ) × ( v = 1 n 1 | Δ v ( a ˆ n v ) | ) k 1 = O ( 1 ) n = 2 m + 1 ( φ n p n P n ) k 1 ( v = 1 n 1 | Δ v ( a ˆ n v ) | | λ v | k | s v | k ) = O ( 1 ) v = 1 m | λ v | k | s v | k n = v + 1 m + 1 ( φ n p n P n ) k 1 | Δ v ( a ˆ n v ) | = O ( 1 ) v = 1 m | λ v | k | s v | k ( φ v p v P v ) k 1 n = v + 1 m + 1 | Δ v ( a ˆ n v ) | = O ( 1 ) v = 1 m | λ v | k 1 | λ v | | s v | k ( φ v p v P v ) k 1 ( p v P v ) = O ( 1 ) v = 1 m | λ v | φ v k 1 ( p v P v ) k | s v | k = O ( 1 ) v = 1 m 1 Δ | λ v | r = 1 v φ r k 1 ( p r P r ) k | s r | k + O ( 1 ) | λ m | v = 1 m φ v k 1 ( p v P v ) k | s v | k = O ( 1 ) v = 1 m 1 | Δ λ v | X v + O ( 1 ) | λ m | X m = O ( 1 ) v = 1 m 1 β v X v + O ( 1 ) | λ m | X m = O ( 1 ) as  m ,
by virtue of the hypotheses of the theorem and the lemma. Again, applying Hölder’s inequality and using the fact that v β v = O ( 1 X v ) = O ( 1 ) by (23), we get that
n = 2 m + 1 φ n k 1 | T n ( 2 ) | k = O ( 1 ) n = 2 m + 1 φ n k 1 ( v = 1 n 1 | a ˆ n , v + 1 | | Δ λ v | | s v | ) k = O ( 1 ) n = 2 m + 1 φ n k 1 ( v = 1 n 1 | a ˆ n , v + 1 | β v | s v | k ) × ( v = 1 n 1 | a ˆ n , v + 1 | β v ) k 1 = O ( 1 ) n = 2 m + 1 φ n k 1 ( v = 1 n 1 | a ˆ n , v + 1 | β v | s v | k ) × ( v = 1 n 1 v | Δ v ( a ˆ n v ) | β v ) k 1 = O ( 1 ) n = 2 m + 1 ( φ n p n P n ) k 1 ( v = 1 n 1 v | Δ v ( a ˆ n v ) | β v | s v | k ) = O ( 1 ) v = 1 m v β v | s v | k n = v + 1 m + 1 ( φ n p n P n ) k 1 | Δ v ( a ˆ n v ) | = O ( 1 ) v = 1 m v β v | s v | k ( φ v p v P v ) k 1 n = v + 1 m + 1 | Δ v ( a ˆ n v ) | = O ( 1 ) v = 1 m v β v | s v | k ( φ v p v P v ) k 1 ( p v P v ) = O ( 1 ) v = 1 m 1 Δ ( v β v ) r = 1 v φ r k 1 ( p r P r ) k | s r | k + O ( 1 ) m β m v = 1 m φ v k 1 ( p v P v ) k | s v | k = O ( 1 ) v = 1 m 1 | Δ ( v β v ) | X v + O ( 1 ) m β m X m = O ( 1 ) v = 1 m 1 v | Δ β v | X v + O ( 1 ) v = 1 m 1 β v + 1 X v + 1 + O ( 1 ) m β m X m = O ( 1 ) as  m ,
by virtue of the hypotheses of the theorem and the lemma. Finally, as in T n ( 1 ) , we have that
n = 1 m φ n k 1 | T n ( 3 ) | k = O ( 1 ) n = 1 m φ n k 1 | a n n λ n s n | k = O ( 1 ) n = 1 m | λ n | φ n k 1 ( p n P n ) k | s n | k = O ( 1 ) as  m .

This completes the proof of the theorem. If we take φ n = P n p n , then we get a result concerning the | A , p n | k summability factors. If we take a n v = p v P n , then we have another result dealing with | N ¯ , p n , φ n | k summability. If we take a n v = p v P n and p n = 1 for all values of n, then we get a result dealing with | C , 1 , φ n | k summability. If we take φ n = n , a n v = p v P n and p n = 1 for all values of n, then we get a result for | C , 1 | k summability. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Erciyes University, Kayseri, 38039, Turkey

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