Journal of Inequalities in Pure and
Applied Mathematics
AN EXTENSION OF RESULTS OF A. MCD. MERCER
AND I. GAVREA
volume 6, issue 4, article 107,
2005.
MAREK NIEZGODA
Department of Applied Mathematics
Agricultural University of Lublin
P.O. Box 158, Akademicka 13
PL-20-950 Lublin, Poland.
Received 19 July, 2005;
accepted 20 September, 2005.
Communicated by: P.S. Bullen
EMail: marek.niezgoda@ar.lublin.pl
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this note we extend recent results of A. McD. Mercer and I. Gavrea on convex
sequences to other classes of sequences.
2000 Mathematics Subject Classification: Primary: 26D15,12E5; Secondary
26A51,39A70.
Key words: Convex sequence, Polynomial, Convex cone, Dual cone, Farkas lemma,
q-class of sequences, Shift operator, Difference operator, Convex sequence of order r.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Basic Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4
Applications for Convex Sequences of Order r . . . . . . . . . . . . 10
References
An Extension of Results of A.
McD. Mercer and I. Gavrea
Marek Niezgoda
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1.
Introduction
The following result is valid [1, 2]. Let a = (a0 , a1 , . . . , an ) be a real sequence.
The inequality
(1.1)
n
X
ak u k ≥ 0
k=0
holds for every convex sequence u = (u0 , u1 , . . . , un ) if and only if the polynomial
n
X
ak x k
Pa (x) :=
An Extension of Results of A.
McD. Mercer and I. Gavrea
Marek Niezgoda
k=0
has x = 1 as a double root and the coefficients ck (k = 0, 1, . . . , n − 2) of the
polynomial
n−2
X
Pa (x)
=
ck xk
(x − 1)2
k=0
are non-negative. The sufficiency and necessity of this result are due, respectively, to A. McD. Mercer [2] and to I. Gavrea [1].
The purpose of this note is to extend the above result to other classes of
sequences u.
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2.
Basic Lemma
A convex cone is a non-empty set C ⊂ Rn+1 such that αC + βC ⊂ C for all
non-negative scalars α and β. We say that a convex cone C is generated by a
set V ⊂ C, and write C = cone V , if every vector in C can be expressed as a
non-negative linear combination of a finite number of vectors in V .
Let h·, ·i stand for the standard inner product on Rn+1 . The dual cone of C
is the cone defined by
dual C := {u ∈ Rn+1 : hu, vi ≥ 0 for all v ∈ C}.
It is well-known that
(2.1)
An Extension of Results of A.
McD. Mercer and I. Gavrea
Marek Niezgoda
dual dual C = C
n+1
for any closed convex cone C ⊂ R
(cf. [3, Theorem 14.1, p. 121]). The
result below is a key fact in our considerations. It is a consequence of (2.1) for
a finitely generated cone C = cone {v0 , v1 , . . . , vp }.
Lemma 2.1 (Farkas lemma). Let v, v0 , v1 , . . . , vp be vectors in Rn+1 . The following two statements are equivalent:
(i) The inequality hu, vi ≥ 0 holds for all u ∈ Rn+1 such that hu, vi i ≥ 0,
i = 0, 1, . . . , p.
(ii) There exist non-negative scalars ci , i = 0, 1, . . . , p, such that
v = c0 v0 + c1 v1 + · · · + cp vp .
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3.
Main Result
Given a sequence q = (q0 , q1 , . . . , qr ) ∈ Rr+1 with 0 ≤ r ≤ n, we define
(3.1) vi := (0, . . . , 0, q0 , q1 , . . . , qr , 0, . . . , 0) = S i v0 ∈ Rn+1
| {z }
i times
for i = 0, 1, . . . , n − r.
Here S is the shift operator from Rn+1 to Rn+1 defined by
(3.2)
An Extension of Results of A.
McD. Mercer and I. Gavrea
S(z0 , z1 , . . . , zn ) := (0, z0 , z1 , . . . , zn−1 ).
Marek Niezgoda
A sequence u = (u0 , u1 , . . . , un ) ∈ R
(3.3)
n+1
is said to be of q-class, if
hu, vi i ≥ 0 for all i = 0, 1, . . . , n − r.
In other words, the q-class consists of all vectors of the cone
D := dual cone {v0 , v1 , . . . , vn−r }.
(3.4)
Example 3.1.
(a). Set r = 0, q0 = 1 and
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vi = (0, . . . , 0, 1, 0, . . . , 0) ∈ R
| {z }
i times
n+1
for i = 0, 1, . . . , n.
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Then (3.3) reduces to
J. Ineq. Pure and Appl. Math. 6(4) Art. 107, 2005
ui ≥ 0 for i = 0, 1, . . . , n.
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Thus D is the class of non-negative sequences.
(b). Put r = 1, q0 = −1 and q1 = 1, and denote
vi = (0, . . . , 0, −1, 1, 0, . . . , 0) ∈ Rn+1 for i = 0, 1, . . . , n − 1.
| {z }
i times
Then (3.3) gives
ui ≤ ui+1 for i = 0, 1, . . . , n − 1,
An Extension of Results of A.
McD. Mercer and I. Gavrea
which means that D is the class of non-decreasing sequences.
(c). Consider r = 2, q0 = 1, q1 = −2, q2 = 1 and
Marek Niezgoda
vi = (0, . . . , 0, 1, −2, 1, 0, . . . , 0) ∈ R
| {z }
i times
n+1
for i = 0, 1, . . . , n − 2.
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In this case, (3.3) is equivalent to
ui+1 ≤
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ui + ui+2
for i = 0, 1, . . . , n − 2.
2
This says that u is a convex sequence. Therefore D is the class of convex sequences.
n+1
Theorem 3.1. Let a = (a0 , a1 , . . . , an ) ∈ R
and q = (q0 , q1 , . . . , qr ) ∈ R
be given with 0 ≤ r ≤ n. Then the inequality
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(3.5)
n
X
k=0
ak u k ≥ 0
J. Ineq. Pure and Appl. Math. 6(4) Art. 107, 2005
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holds for every sequence u = (u0 , u1 , . . . , un ) of q-class if and only if the polynomial
n
X
Pa (x) :=
ak x k
k=0
is divisible by the polynomial
Pq (x) :=
r
X
qk x k ,
k=0
An Extension of Results of A.
McD. Mercer and I. Gavrea
and the coefficients ck (k = 0, 1, . . . , n − r) of the polynomial
Marek Niezgoda
n−r
Pa (x) X
=
ck xk
Pq (x) k=0
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are non-negative.
Proof. The map ϕ that assigns to each sequence b = (b0 , b1 , . . . , bm ) the polynomial
m
X
ϕ(b) := Pb (x) :=
bk x k
k=0
m+1
is a one-to-one linear map from R
to the space of polynomials of degree at
−1
most m. Also, ψ := ϕ is a one-to-one linear map. It is not difficult to check
that
ψ(xk Pb (x)) = S k ψ(Pb (x)).
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Therefore, for any polynomial
Pc (x) := c0 + c1 x + · · · + cn−r xn−r ,
we have
ψ(Pc (x)Pq (x)) = c0 S 0 v0 + c1 S 1 v0 + · · · + cn−r S n−r v0
= c0 v0 + c1 v1 + · · · + cn−r vn−r ,
where vi are given by (3.1). In other words,
(3.6)
Pc (x)Pq (x) = ϕ(c0 v0 + c1 v1 + · · · + cn−r vn−r )
for any c = (c0 , c1 , . . . , cn−r ).
An Extension of Results of A.
McD. Mercer and I. Gavrea
Marek Niezgoda
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We are now in a position to show that the following statements are mutually
equivalent:
(i) Inequality (3.5) holds for every u of q-class.
(ii) ha, ui ≥ 0 for every u ∈ dual cone {v0 , v1 , . . . , vn−r }.
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(iii) There exist non-negative scalars c0 , c1 , . . . , cn−r such that a = c0 v0 +c1 v1 +
· · · + cn−r vn−r .
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(iv) There exist non-negative scalars c0 , c1 , . . . , cn−r such that Pa (x) = (c0 +
c1 x + · · · + cn−r xn−r )Pq (x).
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In fact, (ii) is an easy reformulation of (i) (see (3.4)). That (ii) and (iii) are
equivalent is a direct consequence of Farkas lemma (see Lemma 2.1). We now
show the validity of the implication (iii) ⇒ (iv). By (iii) and (3.6), we have
Pa (x) = ϕ(a) = ϕ(c0 v0 + c1 v1 + · · · + cn−r vn−r ) = Pc (x)Pq (x)
for certain scalars ck ≥ 0, k = 0, 1, . . . , n − r. Thus (iv) is proved.
To prove the implication (iv) ⇒ (iii) assume (iv) holds, that is Pa (x) =
Pc (x)Pq (x) with ck ≥ 0, k = 0, 1, . . . , n − r. Then by (3.6),
a = ψ(Pa (x)) = ψ(Pc (x)Pq (x))
= ψϕ(c0 v0 + c1 v1 + · · · + cn−r vn−r )
= c0 v0 + c1 v1 + · · · + cn−r vn−r .
This completes the proof of Theorem 3.1.
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McD. Mercer and I. Gavrea
Marek Niezgoda
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Applications for Convex Sequences of Order r
4.
In this section we study special types of sequences related to difference calculus
and generalized convex sequences.
We introduce the difference operator on sequences z = (z0 , z1 , . . . , zm ) by
∆z := (z1 − z0 , z2 − z1 , . . . , zm − zm−1 ).
Notice that ∆ = ∆m acts from Rm+1 to Rm . We define
0
r
∆ z := z and ∆ z := ∆m−r+1 · · · ∆m−1 ∆m z for r = 1, 2, . . . , m.
An Extension of Results of A.
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Marek Niezgoda
A sequence u ∈ R
n+1
is said to be convex of order r (in short, r-convex), if
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∆r u ≥ 0.
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The last inequality is meant in the componentwise sense in Rn+1−r , that is
(4.1)
r
h∆ u, ei i ≥ 0 for i = 0, 1, . . . , n − r,
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where
n+1−r
.
ei := (0, . . . , 0, 1, 0, . . . , 0) ∈ R
| {z }
i times
In order to relate the r-convex sequences to the q-class of Section 3, observe
that (4.1) amounts to
hu, (∆r )T ei i ≥ 0 for i = 0, 1, . . . , n − r,
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where (·)T denotes the transpose. By a standard induction argument, we get
(∆r )T ei = S i v0 for i = 0, 1, . . . , n − r,
where S is the shift operator from Rn+1 to Rn+1 given by (3.2), and
(4.2)
v0 := (q, 0, . . . , 0) ∈ Rn+1 and q := (q0 , q1 , . . . , qr )
r
with qj :=
(−1)r−j .
j
An Extension of Results of A.
McD. Mercer and I. Gavrea
As in (3.1), we set
vi := S i v0 for i = 0, 1, . . . , n − r.
In summary, the r-convex sequences form the q-class for q given by (4.2).
For example, the class of r-convex sequences for r = 0 (resp. r = 1, r = 2)
is the class of non-negative (resp. non-decreasing, convex) sequences in Rn+1
(cf. Example 3.1).
By virtue of (4.2) we get
Pq (x) =
r
X
qk xk = (x − 1)r .
k=0
Therefore we obtain from Theorem 3.1
Corollary 4.1. Let a = (a0 , a1 , . . . , an ) ∈ Rn+1 be given with 0 ≤ r ≤ n. Then
the inequality
(4.3)
n
X
k=0
ak uk ≥ 0
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holds for every r-convex sequence u = (u0 , u1 , . . . , un ) if and only if the polynomial
n
X
Pa (x) =
ak x k
k=0
has x = 1 as a root of multiplicity at least r, and the coefficients ck (k =
0, 1, . . . , n − r) of the polynomial
n−r
X
Pa (x)
=
ck xk
r
(x − 1)
k=0
are non-negative.
Corollary 4.1 extends the mentioned results of A. McD. Mercer and I. Gavrea
from r = 2 to an arbitrary 0 ≤ r ≤ n.
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McD. Mercer and I. Gavrea
Marek Niezgoda
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References
[1] I. GAVREA, Some remarks on a paper by A. McD. Mercer, J. Inequal. Pure
Appl. Math., 6(1) (2005), Art. 26. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=495]
[2] A. McD. MERCER, Polynomials and convex sequence inequalities, J. Inequal. Pure Appl. Math., 6(1) (2005), Art. 8. [ONLINE: http://jipam.
vu.edu.au/article.php?sid=477]
[3] R.T. ROCKAFELLAR, Convex Analysis, Princeton University Press,
Princeton 1970.
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McD. Mercer and I. Gavrea
Marek Niezgoda
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