Nonnegative Infinite Matrices that Preserve -Convexity of Sequences

Abstract

This paper deals with matrix transformations that preserve the -convexity of sequences. The main result gives the necessary and sufficient conditions for a nonnegative infinite matrix to preserve the -convexity of sequences. Further, we give examples of such matrices for different values of and .

1. Introduction

If , , then the sequence of real numbers is said to be -convex iffor The operator generates the second-order difference when Several authors [1–3] have proved various results on the convex sequences defined by . Other authors [4, 5] have studied the classes of sequences satisfying . Also, the necessary and sufficient conditions for a sequence to be a -convex sequence can be found in [6]. Moreover, some inequalities on -convex sequences are given in [7, 8].

In [9–11], the authors discuss the matrix transformations that preserve -convexity of sequences in the case of a lower triangular matrix with a particular type of matrix transformation. But the question of a general infinite matrix preserving -convexity has not been considered anywhere in the literature. This paper deals with the necessary and sufficient conditions for a nonnegative infinite matrix to preserve -convexity in both settings when and .

2. Preliminaries

For any given sequence , we can find a corresponding sequence such thatand, for ,which implies that can be represented by and, for ,As a consequence, we get the following lemma. A variation of this lemma can be found in [6].

Lemma 1. If the sequence is given by representation (5), then . Thus, the sequence is -convex if and only if for .

Proof. It suffices to show that for . Using (5), On the right side, we see that the coefficient of , and the coefficient of for . Thus,Hence, we have the previous lemma.
Also, in (5), the representation of in terms of can be written as follows:Now, we give below some definitions. Let be a nonnegative infinite matrix defining a sequence to sequence transformation byThen, we define the matrices and asInterchanging the order of summation, we get, for each , and ,Furthermore, for , In order for the matrix to be well-defined, we need the matrix to satisfy certain conditions which will depend on the values of and .
(I) When , due to symmetry of and in the definition of , it is sufficient to consider the following cases:Case (a). For , we require the matrix to satisfy that, for each , Thus, using (11) and , we have Thus, is well-defined.
Case (b). For , we require the matrix to satisfy that, for each ,Then using (11), we have Thus, is well-defined.
For the cases (c), (d), and (e), we require the matrix to satisfy that, for each ,Case (c). When , we have, as in the case (b), Thus, is well-defined.
Case (d). When , from (11), Since , using (18), we have . Therefore, Thus is well-defined.
Case (e). When , we can assume without loss of generality that
Proceeding as in case (d), we see that is well-defined in this case also.
(II) When , we consider the following cases:Case (f). For , we require the matrix to satisfy that, for each , Then, using (11), we have Thus, is well-defined.
Case (g). When , -convexity reduces to the well-known second-order convexity , which has been investigated in detail in [3].
Case (h). For , we require the matrix to satisfy that, for each , Then, using (11), we have Thus, is well-defined.

3. Main Results

In this section, we prove the necessary and sufficient conditions for a nonnegative infinite matrix to transform a -convex sequence into a -convex sequence showing that each column of the corresponding matrix is a -convex sequence.

First, we consider the values of and , where results in the cases listed in (13).

Theorem 2. For , a nonnegative infinite matrix satisfying (14), (16), or (18), corresponding to the cases listed in (13), preserves -convexity of sequences if and only if, for ,(i)(ii)(iii) for where the matrix is defined by

Proof. First, we prove a result on the transformed sequence of any -convex sequence . Now, we have, from (8),where for by Lemma 1. Then, the th term of the transformed sequence is Interchanging the order of summation, From (11), we haveThen, for , Thus, for any -convex sequence , Now, to prove the sufficiency of the conditions given in the theorem, assume that (i), (ii), and (iii) are true. Then, by (33), Thus, the sequence is also -convex.
Conversely, assume that the matrix preserves -convexity of the sequences. Suppose that the condition (i) fails to hold. Then there exists an integer such that Consider the following sequence: Then is a -convex sequence because, using (2) and Lemma 1, and, for , Thus, from (33), for the transformed sequence , which contradicts that the transformed sequence must be -convex.
Next, suppose that the condition (ii) is not true. This case can be settled by a similar argument by considering the following sequence: which implies thatNow, suppose that the condition (iii) is not true. Then there exists an integer such that the th column-sequence of the matrix is not -convex. That is, for some , Now, consider the following sequence: Then, is a -convex sequence, because, using (2) and Lemma 1, we getand, for , But, from (33), which again contradicts that is a -convex sequence. This completes the proof.

Theorem 2 generalizes the necessary and sufficient conditions given in [9, Theorem  2, p. 8] in the case of and with .

Next, we consider the values of and where results in the cases listed in (22).

Theorem 3. For , a nonnegative infinite matrix satisfying (23) or (25), corresponding to the cases listed in (22), preserves -convexity of sequences if and only if, for , (i)(ii)(iii) for ,where the matrix is defined by

Proof. First we prove a result on the transformed sequence of any -convex sequence . Now, we have, from (8), where for by Lemma 1. Then, the th term of the transformed sequence is Interchanging the order of summation, From (11), we have Then, for , Thus, for any -convex sequence , Now, to prove the sufficiency of the conditions given in the theorem, assume that (i), (ii), and (iii) are true. Then by (53), Thus, the sequence is also -convex.
Conversely, assume that the matrix preserves -convexity of sequences.
Suppose that the condition (i) fails to hold. Then there exists an integer such that Consider the following sequence: It is easy to see, using (2) and Lemma 1, that is a -convex sequence with Thus, from (53), for the transformed sequence , which contradicts that must be -convex.
Next, suppose that the condition (ii) is not true. This case can be settled by a similar argument by considering the following sequence: which implies that Now, suppose that the condition (iii) is not true. Then there exists an integer such that the th column-sequence of the matrix is not -convex. That is, for some , Consider the -convex sequence: We see that, as in the proof of Theorem 2,which contradicts that is a -convex sequence.
We see that the result on the convexity of sequences given in [3, p. 331] is a particular case of Theorem 3 when . Also, this theorem generalizes the necessary and sufficient conditions for a triangular matrix given in [9, p. 4].

4. Examples

We give below examples of -convexity preserving matrices for each of the cases (a) through (h) given in (13) and (22).

Example for Case (a). Considering , and , we can assume, without loss of generality, that . Let the matrix be defined by Then, for each , Thus, by (14), is well-defined for and . The matrix satisfies the three conditions of Theorem 2 because, for , using (12), in which Therefore, the matrix preserves -convexity of sequences.

Example for Case (b). Considering , , let the matrix be defined by Then, for each ,Thus, by (16), is well-defined for and . The matrix satisfies the three conditions of Theorem 2 because, for , using (12), in which Therefore, the matrix preserves -convexity of sequences.

Example for Case (c). Considering , let matrix be defined by Then, for each , Thus, by (18), is well-defined for and . The matrix satisfies the three conditions of Theorem 2 because, for , as in the previous example (b), Therefore, the matrix preserves -convexity of sequences.

Example for Case (d). Considering , let matrix be defined by Then, for each , Thus, by (18), is well-defined for and . The matrix satisfies the three conditions of Theorem 2 because, for , using (12), in which Therefore, the matrix preserves -convexity of sequences.

Example for Case (e). Considering and , we can assume, without loss of generality, that . Let the matrix be defined byThen, for each , Thus, by (18), is well-defined for and . The matrix satisfies the three conditions of Theorem 2 because, for , as in the previous example (d), Therefore, the matrix preserves -convexity of sequences.

Example for Case (f). Considering , let the matrix be defined by Then, for each , Thus, by (23), is well-defined for and . The matrix satisfies the three conditions of Theorem 3 because, for , using (12), in which Therefore, the matrix preserves -convexity of sequences.

Examples for Case (g). They can be found in [3], since is the same as the second-order convexity .

Example for Case (h). Considering , let the matrix be defined by Therefore, for each , Thus, by (23), is well-defined for and . The matrix satisfies the three conditions of Theorem 3 because, for , using (12), in which Therefore, the matrix preserves the convexity of sequences.

We conclude this paper by giving an example of an infinite matrix which does not preserve -convexity of sequences.

It is interesting to notice that the Borel matrix preserves the -convexity of sequences [3, p. 336], but it does not preserve -convexity when .

The Borel matrix is defined by Then, for each ,Thus, for each of the cases, and , we see that (23) and (25) are satisfied and hence is well-defined for and

From (11),Therefore, which implies that since when . Thus, the condition (i) of Theorem 3 fails in the case of Borel matrix.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.