AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi 10.1155/2017/9167069 9167069 Research Article Nonnegative Infinite Matrices that Preserve (p,q)-Convexity of Sequences http://orcid.org/0000-0003-4386-8611 Selvaraj Chikkanna R. 1 Selvaraj Suguna 1 Banas Jozef Penn State University-Shenango 147 Shenango Avenue Sharon PA 16146 USA psu.edu 2017 252017 2017 27 12 2016 11 04 2017 252017 2017 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.

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

1. Introduction

If p>0, q>0, then the sequence {xn} of real numbers is said to be (p,q)-convex if(1)Δp,qxn=xn-p+qxn-1+pqxn-20for n2. The operator Δp,q generates the second-order difference Δ2 when p=q=1. Several authors  have proved various results on the convex sequences defined by Δ2xn0. Other authors [4, 5] have studied the classes of sequences satisfying Δ1,q(xn)0. Also, the necessary and sufficient conditions for a sequence to be a (p,q)-convex sequence can be found in . Moreover, some inequalities on (p,q)-convex sequences are given in [7, 8].

In , the authors discuss the matrix transformations that preserve (p,q)-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 (p,q)-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 (p,q)-convexity in both settings when pq and p=q.

2. Preliminaries

For any given sequence {xn}, we can find a corresponding sequence {ck} such that(2)c0=x0,c1=x1-p+qc0and, for k2,(3)ck=xk-i=0k-1pk-i+pk-i-1q++pqk-i-1+qk-ici,which implies that {xn} can be represented by(4)x0=c0,x1=c1+p+qc0, and, for n2,(5)xn=cn+p+qcn-1+p2+pq+q2cn-2++pn+pn-1q++pqn-1+qnc0=cn+i=1npi+pi-1q++pqn-i+qicn-i.As a consequence, we get the following lemma. A variation of this lemma can be found in .

Lemma 1.

If the sequence {xn} is given by representation (5), then Δp,q(xn)=cn. Thus, the sequence {xn} is (p,q)-convex if and only if cn0 for n2.

Proof.

It suffices to show that Δp,q(xn)=xn-(p+q)xn-1+pqxn-2=cn for n2. Using (5), (6)Δp,qxn=cn+p+qcn-1+p2+pq+q2cn-2++0npn-kqkc0-p+qcn-1+p+qcn-2+p2+pq+q2cn-3++0n-1pn-k-1qkc0+pqcn-2+p+qcn-3+p2+pq+q2cn-4++0n-2pn-k-2qkc0.On the right side, we see that the coefficient of cn=1, and the coefficient of cn-r=0 for r=1,2,,n. Thus,(7)Δp,qxn=cnfor  n2.Hence, we have the previous lemma.

Also, in (5), the representation of xn in terms of cn can be written as follows:(8)xn=cn+i=0n-1pn-i+pn-i-1q++qn-ici=cn+i=0n-1pn-i+1-qn-i+1p-qci,if  pqcn+i=0n-1n-i+1pn-ici,if  p=q=i=0npn-i+1-qn-i+1p-qci,if  pqi=0nn-i+1pn-iciif  p=q.Now, we give below some definitions. Let A=[an,k] be a nonnegative infinite matrix defining a sequence to sequence transformation by(9)Axn=k=0an,kxk.Then, we define the matrices [αn,k] and [βn,k] as(10)αn,k=j=kpj-kan,j=an,k+pan,k+1+p2an,k+2+,βn,i=k=iqk-iαn,k=αn,i+qαn,i+1+q2αn,i+2+=k=iqk-ij=kpj-kan,j.Interchanging the order of summation, we get, for each n=0,1,2,, and i=0,1,2,,(11)βn,i=j=ik=ijqk-ipj-kan,j=j=ipj-i+qpj-i-1+q2pj-i-2++qj-ian,j=j=ipj-i+1-qj-i+1p-qan,j,if  pqj=ij-i+1pj-ian,j,if  p=q.Furthermore, for n2,(12)Δp,qβn,i=βn,i-p+qβn-1,i+pqβn-2,i=j=ipj-i+1-qj-i+1p-qΔp,qan,j,if  pqj=ij-i+1pj-iΔp,qan,j,if  p=q. In order for the matrix [βn,i] to be well-defined, we need the matrix [an,k] to satisfy certain conditions which will depend on the values of p and q.

(I) When p q , due to symmetry of p and q in the definition of βn,i, it is sufficient to consider the following cases:(13)a0<p,q<1b0<p<1,  q=1cp>1,  q=1dp>1,  0<q<1ep,q>1

Case (a). For 0<p,q<1, we require the matrix A to satisfy that, for each n,(14)k=1kan,k<. Thus, using (11) and p,q<1, we have (15)βn,i=j=ipj-i+qpj-i-1++qj-ian,j<j=ij-i+1an,j=j=ij-ian,j+j=ian,j<by  14.Thus, βn,i is well-defined.

Case (b). For 0<p<1,q=1, we require the matrix A to satisfy that, for each n,(16)k=0an,k<.Then using (11), we have (17)βn,i=j=i1-pj-i+11-pan,j=11-p1-pan,i+1-p2an,i+1+<11-pan,i+an,i+1+,  since  0<p<1<by  16.Thus, βn,i is well-defined.

For the cases (c), (d), and (e), we require the matrix A to satisfy that, for each n,(18)k=0pkan,k<.

Case (c). When p>1,q=1, we have, as in the case (b), (19)βn,i=j=ipj-i+1-1p-1an,j=pp-1j=ipj-i-1pan,j<pp-1j=ipj-ian,jpp-1j=ipjan,j<by  18. Thus, βn,i is well-defined.

Case (d). When p>1,0<q<1, from (11),(20)βn,i=1p-qj=ipj-i+1-qj-i+1an,j. Since q<p, using (18), we have j=iqj-ian,j<j=ipj-ian,j<. Therefore, (21)βn,i=pp-qj=ipj-ian,j-qp-qj=iqj-ian,j<. Thus βn,i is well-defined.

Case (e). When p,q>1, we can assume without loss of generality that p>q.

Proceeding as in case (d), we see that βn,i is well-defined in this case also.

(II) When p = q , we consider the following cases:(22)f0<p<1gp=1hp>1.

Case (f). For 0<p<1, we require the matrix A to satisfy that, for each n,(23)k=1kan,k<. Then, using (11), we have (24)βn,i=j=ij-i+1pj-ian,j<j=ij-i+1an,j=j=ij-ian,j+j=ian,j<by  23. Thus, βn,i is well-defined.

Case (g). When p=1, Δp,q-convexity reduces to the well-known second-order convexity Δ2, which has been investigated in detail in .

Case (h). For p>1, we require the matrix A to satisfy that, for each n,(25)k=1kpkan,k<. Then, using (11), we have (26)βn,i=j=ij-i+1pj-ian,jj=ij-ipj-ian,j+j=ipjan,j<by  25.Thus, βn,i is well-defined.

3. Main Results

In this section, we prove the necessary and sufficient conditions for a nonnegative infinite matrix A to transform a (p,q)-convex sequence into a (p,q)-convex sequence showing that each column of the corresponding matrix [βn,k] is a (p,q)-convex sequence.

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

Theorem 2.

For pq, a nonnegative infinite matrix A satisfying (14), (16), or (18), corresponding to the cases listed in (13), preserves (p,q)-convexity of sequences if and only if, for n=2,3,4,,

Δp,qβn,0=0

Δp,qβn,1=0

Δp,qβn,i0 for i2

where the matrix [βn,i] is defined by(27)βn,i=j=ipj-i+1-qj-i+1p-qan,j.

Proof.

First, we prove a result on the transformed sequence of any (p,q)-convex sequence {xn}. Now, we have, from (8),(28)xn=i=0npn-i+1-qn-i+1p-qci,where ci0 for i2 by Lemma 1. Then, the nth term of the transformed sequence is (29)Axn=k=0an,kxk=k=0an,ki=0kpk-i+1-qk-i+1p-qci. Interchanging the order of summation, (30)Axn=i=0cik=ipk-i+1-qk-i+1p-qan,k=c0k=0pk+1-qk+1p-qan,k+c1k=1pk-qkp-qan,k+i=2cik=ipk-i+1-qk-i+1p-qan,k. From (11), we have(31)Axn=c0βn,0+c1βn,1+i=2ciβn,i.Then, for n2, (32)Δp,qAxn=Axn-p+qAxn-1+pqAxn-2=c0βn,0+c1βn,1+i=2ciβn,i-p+qc0βn-1,0+c1βn-1,1+i=2ciβn-1,i+pqc0βn-2,0+c1βn-2,1+i=2ciβn-2,i=c0βn,0-p+qβn-1,0+pqβn-2,0+c1βn,1-p+qβn-1,1+pqβn-2,1+i=2ciβn,i-p+qβn-1,i+pqβn-2,i. Thus, for any (p,q)-convex sequence {xn},(33)Δp,qAxn=c0Δp,qβn,0+c1Δp,qβn,1+i=2ciΔp,qβn,i. Now, to prove the sufficiency of the conditions given in the theorem, assume that (i), (ii), and (iii) are true. Then, by (33),(34)Δp,qAxn0. Thus, the sequence (Ax)n is also (p,q)-convex.

Conversely, assume that the matrix A preserves (p,q)-convexity of the sequences. Suppose that the condition (i) fails to hold. Then there exists an integer N2 such that(35)Δp,qβN,0=L0. Consider the following sequence:(36)u=-L,-p2-q2p-qL,-p3-q3p-qL,. Then {un} is a (p,q)-convex sequence because, using (2) and Lemma 1,(37)c0=u0=-L,c1=u1-p+qc0=0 and, for i2,(38)ci=Δp,qui=ui-p+qui-1+pqui-2=-pi+1-qi+1p-qL+p+qpi-qip-qL-pqpi-1-qi-1p-qL=0. Thus, from (33), for the transformed sequence {(Au)n}, (39)Δp,qAuN=c0Δp,qβN,0+c1Δp,qβN,1+i=2ciΔp,qβN,i=-L2<0,which contradicts that the transformed sequence {(Au)n} must be (p,q)-convex.

Next, suppose that the condition (ii) is not true. This case can be settled by a similar argument by considering the following sequence: (40)v=0,-L,-p2-q2p-qL,-p3-q3p-qL,, which implies that(41)c0=0,c1=-L,ci=0for  i2.Now, suppose that the condition (iii) is not true. Then there exists an integer j2 such that the jth column-sequence of the matrix [βn,k] is not (p,q)-convex. That is, for some N2,(42)Δp,qβN,j=L<0. Now, consider the following sequence: (43)x=0,,0,1,p2-q2p-q,p3-q3p-q,.x0xj-1  xjxj+1Then, {xn} is a (p,q)-convex sequence, because, using (2) and Lemma 1, we get(44)ci=0for  0ij-1;cj=1;cj+1=xj+1-p+qxj+pqxj-1=0;and, for ij+2,(45)ci=Δp,qxi=0as  in  38. But, from (33), (46)Δp,qAxN=c0Δp,qβN,0+c1Δp,qβN,1+i=2ciΔp,qβN,i=cjΔp,qβN,j=L<0,which again contradicts that {Ax} is a (p,q)-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 p=1 and q>0 with q1.

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

Theorem 3.

For p=q, a nonnegative infinite matrix A satisfying (23) or (25), corresponding to the cases listed in (22), preserves (p,q)-convexity of sequences if and only if, for n=2,3,4,,

Δp,p(βn,0)=0

Δp,p(βn,1)=0

Δp,p(βn,i)0 for i=2,3,,

where the matrix [βn,i] is defined by(47)βn,i=j=ij-i+1pj-ian,j.

Proof.

First we prove a result on the transformed sequence of any (p,p)-convex sequence {xn}. Now, we have, from (8),(48)xn=i=0nn-i+1pn-ici, where ci0 for i2 by Lemma 1. Then, the nth term of the transformed sequence is (49)Axn=k=0an,kxk=k=0an,ki=0kk-i+1pk-ici. Interchanging the order of summation, (50)Axn=i=0cik=ik-i+1pk-ian,k=c0k=0k+1pkan,k+c1k=1kpk-1an,k+i=2cik=ik-i+1pk-ian,k. From (11), we have(51)Axn=c0βn,0+c1βn,1+i=2ciβn,i. Then, for n2, (52)Δp,pAxn=Axn-2pAxn-1+p2Axn-2=c0βn,0+c1βn,1+i=2ciβn,i-2pc0βn-1,0+c1βn-1,1+i=2ciβn-1,i+p2c0βn-2,0+c1βn-2,1+i=2ciβn-2,i.Thus, for any (p,p)-convex sequence {xn},(53)Δp,pAxn=c0Δp,pβn,0+c1Δp,pβn,1+i=2ciΔp,pβn,i. Now, to prove the sufficiency of the conditions given in the theorem, assume that (i), (ii), and (iii) are true. Then by (53),(54)Δp,pAxn0. Thus, the sequence (Ax)n is also (p,p)-convex.

Conversely, assume that the matrix A preserves (p,p)-convexity of sequences.

Suppose that the condition (i) fails to hold. Then there exists an integer N2 such that(55)Δp,pβN,0=L0. Consider the following sequence:(56)u=-L,-2pL,-3p2L,. It is easy to see, using (2) and Lemma 1, that u is a (p,p)-convex sequence with(57)c0=uo=-L,ci=0for  i1. Thus, from (53), for the transformed sequence {(Au)n}, (58)Δp,pAuN=c0Δp,pβN,0+c1Δp,pβN,1+i=2ciΔp,pβN,i=-L2<0, which contradicts that {(Au)n} must be (p,p)-convex.

Next, suppose that the condition (ii) is not true. This case can be settled by a similar argument by considering the following sequence: (59)v=0,-L,-2pL,-3p2L,, which implies that(60)c0=0,c1=-L,ci=0for  i2. Now, suppose that the condition (iii) is not true. Then there exists an integer j2 such that the jth column-sequence of the matrix [βn,k] is not (p,p)-convex. That is, for some N2,(61)Δp,pβN,j=L<0. Consider the (p,p)-convex sequence:(62)x=0,,0,1,2p,3p2,.x0xj-1xjxj+1 We see that, as in the proof of Theorem 2,(63)Δp,pAxN=L<0,which contradicts that {Ax} is a (p,p)-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 p=q=1. 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 (p,q)-convexity preserving matrices for each of the cases (a) through (h) given in (13) and (22).

Example for Case (a). Considering 0<p,q<1, and pq, we can assume, without loss of generality, that p<q. Let the matrix A=[an,k] be defined by (64)an,k=pn,if  k=0,pnqkk,if  k1.Then, for each n,(65)k=1kan,k=k=1pnqk=pnq1-q<. Thus, by (14), βn,i is well-defined for n=0,1,2, and i=0,1,2,. The matrix A satisfies the three conditions of Theorem 2 because, for n2, using (12),(66)Δp,qβn,i=j=ipj-i+1-qj-i+1p-qΔp,qan,j, in which (67)Δp,qan,j=an,j-p+qan-1,j+pqan-2,j=pn-p+qpn-1+pqpn-2,if  j=0,qjjpn-p+qpn-1+pqpn-2,if  j1=0. Therefore, the matrix A preserves (p,q)-convexity of sequences.

Example for Case (b). Considering 0<p<1, q=1, let the matrix A=[an,k] be defined by(68)an,k=pk. Then, for each n,(69)k=0an,k=k=0pk=11-p<.Thus, by (16), βn,i is well-defined for n=0,1,2, and i=0,1,2,. The matrix A satisfies the three conditions of Theorem 2 because, for n2, using (12),(70)Δp,1βn,i=j=ipj-i+1-1p-1Δp,1an,j, in which(71)Δp,1an,j=an,j-p+1an-1,j+pan-2,j=pj-p+1pj+pj+1=0. Therefore, the matrix A preserves (p,1)-convexity of sequences.

Example for Case (c). Considering p>1,q=1, let matrix A=[an,k] be defined by(72)an,k=1p2k. Then, for each n,(73)k=0pkan,k=k=01pk=11-1/p<. Thus, by (18), βn,i is well-defined for n=0,1,2, and i=0,1,2,. The matrix A satisfies the three conditions of Theorem 2 because, for n2, as in the previous example (b),(74)Δp,1an,j=an,j-p+1an-1,j+pan-2,j=1p2j-p+11p2j+p1p2j=0. Therefore, the matrix A preserves (p,1)-convexity of sequences.

Example for Case (d). Considering p>1,0<q<1, let matrix A=[an,k] be defined by(75)an,k=pn-2k. Then, for each n,(76)k=0pkan,k=k=0pn-k=pn+1p-1<. Thus, by (18), βn,i is well-defined for n=0,1,2, and i=0,1,2,. The matrix A satisfies the three conditions of Theorem 2 because, for n2, using (12),(77)Δp,qβn,i=j=ipj-i+1-qj-i+1p-qΔp,qan,j, in which(78)Δp,qan,j=an,j-p+qan-1,j+pqan-2,j=1p2jpn-p+qpn-1+pqpn-2=0. Therefore, the matrix A preserves (p,q)-convexity of sequences.

Example for Case (e). Considering p,q>1 and pq, we can assume, without loss of generality, that p>q. Let the matrix A=[an,k] be defined by(79)an,k=pn-2kqk.Then, for each n,(80)k=0pkan,k=pnk=0qpk=pn+1p-q<. Thus, by (18), βn,i is well-defined for n=0,1,2, and i=0,1,2,. The matrix A satisfies the three conditions of Theorem 2 because, for n2, as in the previous example (d),(81)Δp,qan,j=an,j-p+qan-1,j+pqan-2,j=qjp2jpn-p+qpn-1+pqpn-2=0. Therefore, the matrix A preserves (p,q)-convexity of sequences.

Example for Case (f). Considering 0<p=q<1, let the matrix A=[an,k] be defined by (82)an,k=pn,if  k=0,pn+kk,if  k1. Then, for each n,(83)k=1kan,k=k=1pn+k=pnk=1pk=pn+111-p<. Thus, by (23), βn,i is well-defined for n=0,1,2, and i=0,1,2,. The matrix A satisfies the three conditions of Theorem 3 because, for n2, using (12),(84)Δp,pβn,i=j=ij-i+1pj-iΔp,pan,j, in which (85)Δp,pan,j=an,j-2pan-1,j+p2an-2,j=pn-2ppn-1+p2pn-2,if  j=0,pjjpn-2ppn-1+p2pn-2,if  j1=0. Therefore, the matrix A preserves (p,p)-convexity of sequences.

Examples for Case (g). They can be found in , since Δ1,1 is the same as the second-order convexity Δ2.

Example for Case (h). Considering p=q>1, let the matrix A=[an,k] be defined by (86)an,k=pnn+2,if  k=0,pn-2kn+2k,if  k1.Therefore, for each n, (87)k=1kpkan,k=k=1pn-kn+2=pnn+2k=11pk=n+2pnp-1<. Thus, by (23), βn,i is well-defined for n=0,1,2, and i=0,1,2,. The matrix A satisfies the three conditions of Theorem 3 because, for n2, using (12),(88)Δp,pβn,i=j=ij-i+1pj-iΔp,pan,j, in which (89)Δp,pan,j=an,j-2pan-1,j+p2an-2,j=pnn+2-2pnn+1+pnnif  j=0,n+2pn-2jj-2n+1pn-2jj+npn-2jj,if  j1.=0. Therefore, the matrix A preserves the convexity of sequences.

We conclude this paper by giving an example of an infinite matrix which does not preserve (p,q)-convexity of sequences.

It is interesting to notice that the Borel matrix preserves the (1,1)-convexity of sequences [3, p. 336], but it does not preserve (p,p)-convexity when p1.

The Borel matrix B=[bn,k] is defined by(90)bn,k=nkenk!. Then, for each n,(91)k=1kbn,k=nenk=1nk-1k-1!=n<,(92)k=1kpkbn,k=npenk=1npk-1k-1!=npenenp<.Thus, for each of the cases, 0<p<1 and p>1, we see that (23) and (25) are satisfied and hence βn,i is well-defined for n=0,1,2, and i=0,1,2.

From (11),(93)βn,i=j=ij-i+1pj-ibn,j.Therefore, (94)βn,0=j=0j+1pjnjenj!=1enj=0jpnjj!+j=0pnjj!=1enpnepn+epn=enp-1pn+1, which implies that (95)Δp,pβn,0=βn,0-2pβn-1,0+p2βn-2,0=enp-1pn+1-2pen-1p-1pn-1+1+p2en-2p-1pn-2+1=enp-1e2ppn+1ep-pe2+2p2eep-pe>0, since ep-pe>0 when p1. 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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