Journal of Inequalities in Pure and
Applied Mathematics
ZERO AND COEFFICIENT INEQUALITIES FOR
HYPERBOLIC POLYNOMIALS
volume 7, issue 1, article 25,
2006.
J. RUBIÓ-MASSEGÚ, J.L. DÍ AZ-BARRERO AND P. RUBIÓ-DÍ AZ
Applied Mathematics III
Universitat Politècnica de Catalunya
Jordi Girona 1-3, C2, 08034 Barcelona, Spain.
Received 26 May, 2005;
accepted 09 December, 2005.
Communicated by: S.S. Dragomir
EMail: josep.rubio@upc.edu
EMail: jose.luis.diaz@upc.edu
EMail: pere.rubio@upc.edu
Abstract
Contents
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Abstract
In this paper using classical inequalities and Cardan-Viète formulae some inequalities involving zeroes and coefficients of hyperbolic polynomials are given.
Furthermore, considering real polynomials whose zeros lie in Re(z) > 0, the
previous results have been extended and new inequalities are obtained.
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
2000 Mathematics Subject Classification: 12D10, 26C10, 26D15.
Key words: Zeroes and coefficients, Inequalities in the complex plane, Inequalities
for polynomials with real zeros, Hyperbolic polynomials.
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
1.
Introduction
The problem of finding relations between the zeroes and coefficients of a polynomial occupies a central role in the theory of equations. The most well known
of such relations are Cardan-Viète’s formulae [1]. Many papers devoted to obtaining inequalities between the zeros and coefficient have been written giving
new bounds or improving the classical known ones ([2], [3], [4]). Furthermore,
inequalities for polynomials with all zeros real also called hyperbolic polynomials, have been fully documented in [5]. In this paper, using some classical
inequalities, several inequalities involving zeros and coefficients of polynomials with real zeros have been obtained and the main result has been extended to
polynomials whose zeros lie in the right half plane.
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
2.
The Inequalities
In what follows some zero and coefficient inequalities involving polynomials
whose zeros are strictly positive real numbers are obtained. We begin with
Pn
k
Theorem 2.1. Let A(x) =
k=0 ak x , an 6= 0, be a hyperbolic polynomial
with all its zeroes x1 , x2 , . . . , xn strictly positive. If α, p and b are strictly positive real numbers such that α < p, then
1 1 −1
n
X
1
α p p − α α p a1
(2.1)
.
1 ≤
1
p
b
a
α
α
0
[x
+
b]
p
k
k=1
p1 n
bα
.
Equality holds when A(x) = an x − p−α
Proof. Let β and a be strictly positive real numbers defined by β = 1 − αp > 0
bα
and a = pβ
> 0. Taking into account that αp + β = 1 and applying the powered
AM-GM inequality, we have for all k, 1 ≤ k ≤ n,
α
α
(2.2)
(xpk ) p aβ ≤ xpk + βa.
p
Inverting the terms in (2.2) yields
1
1
≤ α β,
α p
x + βa
xk a
p k
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
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1 ≤ k ≤ n,
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or equivalently
xpk
α
1
1
≤ · α β.
p
+ α βa
p xk a
Page 4 of 14
J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
Taking into account that αp βa = b and β =
xpk
p−α
,
p
we have
α
1
1
1
≤ · β α
+b
p
xk
bα
pβ
=
α pβ β β 1
·
p bβ αβ xαk
α
α
1−
p−α 1−
α p p( p ) p 1
· α
= ·
α
α
xk
p
b1− p α1− p
α 1− αp
αp p − α
1
=
.
p
b
xαk
Raising to
1
α
both sides of the preceding inequality, yields
1 1 −1
αp p − α α p 1
1
,
1 ≤ k ≤ n.
1 ≤
1
xk
b
[xpk + b] α
pα
Finally, adding up the preceding inequalities, we obtain
1 1 1 −1 n
1 −1
n
X
1
αp p − α α p X 1
α p p − α α p a1
= 1
1 ≤
1
p
b
x
b
a0
α
α
k
[x
+
b]
p
pα
k
k=1
k=1
and (2.1) is proved.
Notice that equality holds in (2.1) if and only if equality holds in (2.2) for
1
1 ≤ k ≤ n. Namely, equality holds when xpk= a, 1 ≤ k ≤
n or xk = a p =
p1 p1
p1 n
bα
bα
bα
= p−α . That is, when A(x) = an x − p−α
, an 6= 0.
pβ
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
When α > p changing α by
1
α
and p by
1
p
into (2.1), we have the following:
Corollary 2.2. If α, p and b are strictly positive real numbers such that α > p,
then
α−p
n
X
1
pp α − p
a1
.
≤ α
1
α
b
a0
p
k=1 [xk + b]α
Multiplying both sides of (2.2) by
p
α
and raising to α1 , we obtain for 1 ≤ k ≤
n,
p α1 β
p α1
a α xk .
≥
βa
α
α
bα
Setting β = 1 − αp , a = pβ
into the preceding expression and, after adding up
the resulting inequalities, we get
xpk +
Corollary 2.3. If α, p and b are strictly positive real numbers such that α < p,
then
α1 − p1
1 n
X
1
pα
b
an−1
p
α
(xk + b) ≥ 1
an
αp p − α
k=1
holds.
Another, immediate consequence of (2.1) is the following.
P
Corollary 2.4. Let A(x) = nk=0 ak xk , an 6= 0, be a hyperbolic polynomial
with all its zeroes x1 , x2 , . . . , xn strictly positive. Then,
n
X
k=1
1
1 a1
≤ 2
n
2
[xk + 2n − 1]
4n a0
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
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holds.
J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
Proof. Setting α = 12 , p = n and b = 2n − 1 into (2.1), we have
n
X
k=1
1
1
1 n 1 2− n
n
−
1
a1
2
≤ 22
n
2
[xk + 2n − 1]
n
2n − 1
a0
1
1
1 n 2− n
1
a1
2
=
2
n
2
a0
1 a1
= 2
.
4n a0
Note that equality holds when xk = 1, 1 ≤ k ≤ n. That is, when A(x) =
an (x − 1)n . This completes the proof.
P
Considering the reverse polynomial A∗ (x) = xn A(1/x) = nk=0 an−k xk ,
we have the following
Theorem 2.5. If α, p and b are strictly positive real numbers such that α < p,
then
1
α1
n X
1
1 an−1
xpk
αp
(2.3)
≤ 1 1 · (p − α) α − p
.
p
x
+
b
a
n
α bp
p
k
k=1
Equality holds when A(x) = an x −
b(p−α)
α
p1 n
, an 6= 0.
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
Proof. Since A∗ (x) has zeros
n
X
k=1
1
h p
1
xk
1
, . . . , x1n ,
x1
+b
i α1
then applying (2.1) to it, we get
1
1 −1
αp
p − α α p an−1
.
≤ 1 ·
b
an
pα
Developing the LHS of the preceding inequality, we have
1
1 −1
α1
n p − α α p an−1
αp
1 X
xpk
≤ 1 ·
,
1
p
1
b
an
pα
b α k=1 b + xk
and rearranging terms, yields
1
α1
1 −1
n X
1
p − α α p an−1
xpk
αp
≤ bα · 1 ·
p
1
an
b
+
x
pα
k
b
k=1
1
=
αp
p
1
α
1
1
1
· b p · (p − α) α − p
an−1
.
an
Finally, replacing b by 1/b in the preceding inequality we get (2.3) as claimed.
Applying Theorem 2.1, equality in (2.3) holds when
p1 !n
α
A∗ (x) = an x −
.
b(p − α)
Taking into account that we have changed b by 1/b, equality will hold if and
only if
1 !n
b(p − α) p
, an 6= 0
A(x) = an x −
α
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
and the proof is completed.
Next, we state and prove the following:
Theorem 2.6. Let A(x) be a hyperbolic polynomial with zeros x1 , x2 , . . . , xn
such that x1 ≤ x2 ≤ · · · ≤ xn . Let α, p and b be strictly positive real numbers
such that α < p. If a < x1 or a > xn , then
1 1 −1
n
X
α p p − α α p P 0 (a)
1
(2.4)
.
1 ≤
1
p
b
P
(a)
α
α
[|x
−
a|
+
b]
p
k
k=1
Equality holds when
"
A(x) = an x − a +
"
A(x) = an x − a −
bα
p−α
bα
p−α
p1 #!n
or
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
p1 #!n
.
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Proof. First, we observe that (2.1) applied to polynomial P (−t) where P (t) has
all its zeros t1 , t2 , . . . , tn negative, yields
1
1 −1
n
X
αp
p − α α p a1
1
,
(2.5)
1 ≤
1 ·
p
b
a
α
α
0
[|t
|
+
b]
p
k
k=1
where equality holds when
P (t) = an t +
bα
p−α
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p1 !n
,
an 6= 0.
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
Now, we consider the hyperbolic polynomial of the statement and assume
that (i) a < x1 or (ii) a > xn . Let B(x) = A(x + a), the zeros of which
are x1 − a, x2 − a, . . . , xn − a. Observe that, they are positive for case (i), and
negative for case (ii). On the other hand, coefficients a0 and a1 of B(x) are
B(0) = A(a) and B 0 (0) = A0 (a) respectively. Applying (2.1) to B(x) in case
(i) or (2.5) in case (ii) we get (2.4).
Finally, we see that equality in (2.4) holds in the case (i) when
p1 !n
bα
B(x) = an x −
,
p−α
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
or equivalently when
"
A(x) = an x − a +
bα
p−α
p1 #!n
.
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In case (ii) we will get equality when
B(x) = an x +
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
bα
p−α
p1 !n
,
an 6= 0.
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That is, when
"
A(x) = an x − a −
bα
p−α
p1 #!n
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Page 10 of 14
and the proof is completed.
J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
Finally, in the sequel we will extend the result obtained in Theorem 2.1 to
real polynomials whose zeros lie in the half plane Re(z) > 0 and they have an
imaginary part “sufficiently small”. This is stated and proved in the following.
P
Theorem 2.7. Let A(z) = nk=0 ak z k be a polynomial with real coefficients
whose zeros z1 , z2 , . . . , zn lie in Re(z) > 0 and suppose that | Im(z)| ≤ r Re(zk ),
1 ≤ k ≤ n for some real r ≥ 0. Let α, p and b be strictly positive real numbers
such that α < p, then
(2.6)
1
n
X
1
k=1
[|zk |p + b] α
1
≤
αp
1
pα
·
p−α
b
α1 − p1
·
√
1+
r2
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
a1
.
a0
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For r > 0, equality holds when n is even and
A(z) =
z2 − √
2
·
1 + r2
bα
p−α
p1
z+
bα
p−α
p2 ! n2
.
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Note that when r = 0 the preceding result reduces to (2.1).
Proof. Setting xk = |zk | and repeating the procedure followed in proving (2.1),
we get
1
1 −1 n
n
X
1
αp
p−α α p X 1
.
1 ≤
1 ·
p
b
|zk |
α
pα
k=1 [|zk | + b]
k=1
P
Next, we will find an upper bound for the sum S = nk=1 |z1k | . Reordering the
zeros of A(z) in the way z1 , z1 , z2 , z2 , . . . , zs , zs , x1 , . . . , xt ,where x1 , . . . , xt are
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
the real zeros (if any), then the preceding sum becomes
s
t
s
t
X
X
X
X
1
1
|zk |
1
S=2
+
=2
.
2 +
|z
|x
|x
|z
|
k|
k|
k|
k
k=1
k=1
k=1
k=1
On the other hand, by Cardan-Viète formulae, we have
X
s t
s
t
X
1
Re zk X 1
a1 X 1
1
− =
+
+
=2
+
.
2
a0
z
z
x
x
|z
|
k
k
k
k
k
k=1
k=1
k=1
k=1
p
√
Taking into account that |zk | = (Re zk )2 + (Im zk )2 ≤ 1 + r2 |Re zk | and
the fact that the zeros of A(z) lie in Re(z) > 0, yields
(2.7)
s
t
X
X
|zk |
1
S=2
2 +
|xk |
|zk |
k=1
k=1
t
s
X
√
|Re zk | X 1
2
≤2 1+r
2 +
|xk |
|z
|
k
k=1
k=1
≤
√
1+
r2
2
s
X
|Re zk |
k=1
=
√
1 + r2
|zk |2
t
X
1
+
|xk |
k=1
Zero and Coefficient
Inequalities for Hyperbolic
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J.L. Díaz-Barrero and
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!
a1
,
a0
from which (2.6) immediately follows.
Next, we will see when equality holds in (2.6). If r > 0, to get equality in
p1
bα
(2.6) we require that (i) all the zeros of A(z) have modulus |zk | = p−α
,
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
because when xk = |zk | the powered GM-AM inequality (2.2) must become
equality, (ii) |Im zk | = r Re zk , 1 ≤ k ≤ s, due to the fact that the inequality
in (2.7) must become equality, and (iii) all the zeros of A(z) must be complex
because the second inequality in (2.7) also must be an equality. Now it is easy
to see that the previous conditions are equivalent to say that n is even and
1
zk = √
1 + r2
bα
p−α
p1
[1 + ri],
1≤k≤
n
.
2
Multiplying the preceding zeros we get that inequality in (2.6) holds when n is
even and
p1
p2 ! n2
2
bα
bα
.
A(z) = z 2 − √
z+
2
p−α
p−α
1+r
This completes the proof.
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
J. Rubió-Massegú,
J.L. Díaz-Barrero and
P. Rubió-Díaz
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J. Ineq. Pure and Appl. Math. 7(1) Art. 25, 2006
References
[1] F. VIÈTE, Opera Mathematica, Leiden, Germany, 1646.
[2] M. MARDEN, Geometry of Polynomials, American Mathematical Society,
Providence, Rhode Island, 1989.
[3] Th.M. RASSIAS, Inequalities for polynomial zeros, 165–202, in Th.M.
Rassias, (ed.), Topics in Polynomials, Kluwer Academic Publisher, Dordrecht, 2000.
[4] Q.I. RAHMAN AND G. SCHMEISSER, Analytic Theory of Polynomials,
Clarendon Press, Oxford 2002.
[5] D. MITRINOVIĆ, G. MILOVANOVIĆ AND Th.M. RASSIAS, Topics in
Polynomials: Extremal Problems, Inequalities and Zeros, World Scientific,
Singapore, 1994.
Zero and Coefficient
Inequalities for Hyperbolic
Polynomials
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J.L. Díaz-Barrero and
P. Rubió-Díaz
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