Abstract and Applied Analysis
Volume 2017 (2017), Article ID 9167069, 9 pages
https://doi.org/10.1155/2017/9167069

Research Article

Nonnegative Infinite Matrices that Preserve -Convexity of Sequences

Chikkanna R. Selvaraj and Suguna Selvaraj

Penn State University-Shenango, 147 Shenango Avenue, Sharon, PA 16146, USA

Correspondence should be addressed to Chikkanna R. Selvaraj; ulf@psu.edu

Received 27 December 2016; Accepted 11 April 2017; Published 2 May 2017

Academic Editor: Jozef Banas

Copyright © 2017 Chikkanna R. Selvaraj and Suguna Selvaraj. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 if

for 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 that

and, 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 by

Then, we define the matrices and as

Interchanging 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 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 (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 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 ,

Now, consider the following sequence:

Then, is a -convex sequence, because, using (2) and Lemma 1, we get

and, 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 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 (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.

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