Journal of Inequalities in Pure and
Applied Mathematics
ASYMPTOTIC EXPANSION OF THE EQUIPOISE CURVE OF A
POLYNOMIAL INEQUALITY
volume 3, issue 5, article 84,
2002.
ROGER B. EGGLETON AND WILLIAM P. GALVIN
Department of Mathematics
Illinois State University
Normal, IL 61790-4520,
USA.
EMail: roger@math.ilstu.edu
School of Mathematical and Physical Sciences
University of Newcastle
Callaghan, NSW 2308
Australia.
EMail: mmwpg@cc.newcastle.edu.au
Received 28 August, 2002;
accepted 8 November, 2002.
Communicated by: H. Gauchman
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
091-02
Quit
Abstract
For any a := (a1 , a2 , . . . , an ) ∈ (R+ )n , define ∆Pa (x, t) := (x + a1 t)(x + a2 t)
· · · (x + an t) − xn and Sa (x, y) := a1 xn−1 + a2 xn−2 y + · · · + an yn−1 . The two homogeneous polynomials ∆Pa (x, t) and tSa (x, y) are comparable in the positive
octant x, y, t ∈ R+ . Recently the authors [2] studied the inequality ∆Pa (x, t) >
tSa (x, y) and its reverse and noted that the boundary between the corresponding regions in the positive octant is fully determined by the equipoise curve
∆Pa (x, 1) = Sa (x, y). In the present paper the asymptotic expansion of the
equipoise curve is shown to exist, and is determined both recursively and explicitly. Several special cases are then examined in detail, including the general
solution when n = 3, where the coefficients involve a type of generalised Catalan number, and the case where a = 1 + δ is a sequence in which each term
is close to 1. A selection of inequalities implied by these results completes the
paper.
2000 Mathematics Subject Classification: 26D05, 26C99, 05A99.
Key words: Polynomial inequality, Catalan numbers.
The first author gratefully acknowledges the hospitality of the School of Mathematical
and Physical Sciences, University of Newcastle, throughout the writing of this paper.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Wider Range of Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Asymptotic Expansion of the Equipoise Curve . . . . . . . . . . . . 9
4
Particular Evaluations of the Asymptotic Expansion . . . . . . . 21
5
Selected Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
1.
Introduction
With any finite real sequence a := (a1 , a2 , . . . , an ) ∈ Rn we associate two
homogeneous polynomials, the product polynomial
n
Y
Pa (x, t) := (x + a1 t)(x + a2 t) · · · (x + an t) =
(x + ar t),
r=1
and the sum polynomial
Sa (x, y) := a1 xn−1 + a2 xn−2 y + · · · + an y n−1 =
n
X
ar xn−r y r−1 .
r=1
As shown in [2], the first difference of the product polynomial
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
∆Pa (x, t) := Pa (x, t) − Pa (x, 0) = Pa (x, t) − xn
Title Page
and t times the sum polynomial are degree n homogeneous polynomials which
are comparable in the positive octant x, y, t ∈ R+ := {r ∈ R : r ≥ 0} when
a ∈ (R+ )n . Clearly they are closely related to the comparison of the product
Πnr=1 (1 + ar ) and the sum Σnr=1 ar . Indeed Weierstrass [4] derived inequalities
equivalent to
Contents
1+
n
X
r=1
n
Y
ar <
(1 + ar ) < Qn
r=1
1
1
Pn
<
1 − r=1 ar
r=1 (1 − ar )
when a ∈ (R+ )n and 0 < Σnr=1 ar < 1, whence the products Πnr=1 (1 + ar ) and
Πnr=1 (1 − ar ) both converge as n → ∞ if Σnr=1 ar converges to a limit strictly
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
less than 1. The first and third of these inequalities correspond to Sa (1, 1) <
∆Pa (1, 1) and −Sa (1, 1) < ∆Pa (1, −1) respectively, while the middle inequality simply follows from 1 − a2r ≤ 1 for 1 ≤ r ≤ n, with strict inequality
for at least one r. The first inequality corresponds to results in [2] at the point
(x, y, t) = (1, 1, 1), but the third corresponds to (1, 1, −1), which is outside the
positive octant, and, although easily proved, it is not covered in [2]. (A more
widely accessible source which closely parallels Weierstrass’s reasoning is [1].)
To summarise the results in [2], let us now suppose that n ≥ 2 and a
is strictly positive, so ar > 0 for 1 ≤ r ≤ n. Then the strict inequality
∆Pa (x, t) > tSa (x, y) holds in a region (the “∆P -region”) of the positive
octant which includes the intersection of the octant with the halfspace y <
x + tm(a), where m(a) := min{ar : 1 ≤ r ≤ n − 1}, and the reverse inequality ∆Pa (x, t) < tSa (x, y) holds in a region (the “S-region”) of the positive
octant which includes its intersection with the halfspace y > x + tM (a), where
M (a) := max{ar : 1 ≤ r ≤ n − 1}. The boundary between the ∆P -region
and the S-region is the equipoise surface
E2 (a) := {(x, y, t) ∈ (R+ )3 : ∆Pa (x, t) = tSa (x, y)}.
The polynomials are homogeneous in a, so for any real t we have
∆Pta (x, 1) = ∆Pa (x, t) and
Sta (x, y) = tSa (x, y),
where ta := (ta1 , ta2 , . . . , tan ) ∈ Rn . Hence for strictly positive a ∈ (R+ )n
with n ≥ 2, it suffices to compare the polynomials in the intersection of the
positive octant with the plane t = 1, so we consider the equipoise curve
E1 (a) := {(x, y) ∈ (R+ )2 : ∆Pa (x, 1) = Sa (x, y)}.
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
This separates the ∆P -region of the positive quadrant x, y ∈ R+ , where
∆Pa (x, 1) > Sa (x, y), from the S-region, where ∆Pa (x, 1) < Sa (x, y). The
equipoise curve lies in the strip
x + m(a) ≤ y ≤ x + M (a)
of the positive quadrant and is asymptotic to y = x + α, where α is a certain
function of a. In fact, if n ≥ 3 the equipoise curve satisfies
y = x + α + βx−1 + O(x−2 )
as x → ∞
where α, β are functions of a explicitly determined in [2]. The equipoise curve
approaches the asymptote from the ∆P -region side if β is negative, and from
the S-region side if β is positive.
Our main purpose in this paper is to extend our understanding of α and β as
functions of a, by determining the subsequent members of an infinite sequence
of coefficients constituting the asymptotic expansion of the equipoise curve for
a. But first we shall show that the properties just summarized hold a little more
generally.
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
2.
Wider Range of Validity
To extend the results of [2] it is convenient to introduce some notation. For
any sequence a := (a1 , a2 , . . . , an ) ∈ Rn and any integer k in the interval
0 ≤ k ≤ n, let
Ak (a) := (a1 , a2 , . . . , ak ) and Ωk (a) := (an−k+1 , . . . , an−1 , an )
be, respectively, the initial and final k-term subsequences of a. Thus An (a) =
Ωn (a) = a and, if ω is the empty sequence, A0 (a) = Ω0 (a) = ω. Also m(a) :=
min{ar : 1 ≤ r ≤ n − 1} = min An−1 (a) and M (a) := max{ar : 1 ≤ r ≤
n − 1} = max An−1 (a). As in [2] we also use Σ(a) := Σnr=1 ar .
First, a simple reformulation of Corollary 2.2 of [2] becomes
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
n
Theorem 2.1. For any finite sequence a ∈ (R+ ) with n ≥ 3, and for all strictly
positive x, y, t ∈ R+ , if An−1 (a) is not constant then for y ≥ x+t max An−1 (a)
we have
0 < tΣ(a)xn−1 < ∆Pa (x, t) < tSa (x, y),
while for y ≤ x + t min An−1 (a) and z := min{x, y} we have
∆Pa (x, t) > tSa (x, y) ≥ tΣ(a)z n−1 > 0.
To investigate the equality ∆Pa (x, 1) = Sa (x, y), in [2] we imposed the
sufficient condition that a ∈ (R+ )n be strictly positive. However, we note that
if x, y are strictly positive then
∂
Sa (x, y) > 0
∂y
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
holds if and only if Ωn−1 (a) 6= 0, where 0 ∈ Rn−1 is the constant sequence with
every term equal to 0. We shall abbreviate this condition by saying “if and only
if Ωn−1 (a) is nonzero”. Then continuity of Sa (x, y) as a function of y ensures
the following broadening of the scope of Lemma 3.1 of [2]:
Theorem 2.2. For any finite sequence a ∈ (R+ )n with n ≥ 2, and strictly
positive x, y ∈ R+ , if Ωn−1 (a) is nonzero then there is a function y0 (x) such
that

< Sa (x, y) if y > y0 (x),



∆Pa (x, 1) = Sa (x, y) if y = y0 (x),



> Sa (x, y) if y < y0 (x).
Furthermore
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
x + min An−1 (a) ≤ y0 (x) ≤ x + max An−1 (a).
Title Page
As in [2], it is convenient now to define two families of sequence functions
Σk , Wk : Rn → R, for any positive integer n and all positive integers k ≤
n. These functions are needed to describe the coefficients in the asymptotic
expansion of y = y0 (x), the equipoise curve for a.
The kth elementary symmetric function Σk of a ∈ Rn is the sum of all
products Πx as x runs through the k-term subsequences x ⊆ a, thus
Contents
Σk (a) := Σ {Πx : x ⊆ a, |x| = k} .
n
In particular, Σ1 (a) = Σnr=1 ar and Σ2 (a) = Σn−1
r=1 Σs=r+1 ar as if n ≥ 2. We
extend the definition by setting Σk (a) = 0 for any integer k > n.
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
The kth binomially-weighted sum Wk of a ∈ Rn is the sequence function
Wk (a) :=
n X
r−1
r=1
k−1
ar .
In particular, W1 (a) = Σnr=1 ar and W2 (a) = Σnr=1 (r − 1)ar if n ≥ 2. Note that
W1 (a) = Σ1 (a) holds for any a. Once again we extend the definition by setting
Wk (a) = 0 for any integer k > n. Now Theorem 2.2 justifies the following
broadening of the scope of Theorem 3.1 and Corollary 3.2 of [2].
Theorem 2.3. For any finite sequence a ∈ (R+ )n with n ≥ 2, and strictly positive x, y ∈ R+ , if Ωn−1 (a) is nonzero then the equality ∆Pa (x, 1) = Sa (x, y)
holds for large x when
y = x + α + βx−1 + O(x−2 ) as x → ∞,
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
where
α := Σ2 (a)/W2 (a)
and
β := (Σ3 (a) − α2 W3 (a))/W2 (a).
Note that β = 0 if n = 2. We will extend Theorem 2.3 in the next section.
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
3.
Asymptotic Expansion of the Equipoise Curve
Let us first establish the existence of the asymptotic expansion of E1 (a) for
suitable a.
Theorem 3.1. For any finite sequence a ∈ (R+ )n with n ≥ 2 and Ωn−1 (a)
nonzero, there is an infinite sequence α := (α1 , α2 , . . .) ∈ R∞ such that the
equipoise curve E1 (a) has asymptotic expansion
!
∞
X
−s
y ∼x 1+
αs x
as x → ∞.
s=1
Proof. By Theorem 2.3, there is an α1 ∈ R such that E1 (a) is y = x + α1 +
O(x−1 ) as x → ∞. Now assume for some positive integer N that there is a
sequence (α1 , α2 , . . . , αN ) ∈ RN such that E1 (a) is
!
N
X
y =x 1+
αs x−s + fN (x),
s=1
with O(fN (x)) = O(x
−N
∆Pa (x, 1) = Sa (x, y)
=
ar xn−r
x 1+
=
r=1
N
X
!
αs x−s
s=1
ar x
n−1
Title Page
Contents
II
I
Go Back
r=1
n
X
Roger B. Eggleton and
William P. Galvin
JJ
J
) as x → ∞. Then
n
X
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
1+
N
X
s=1
!r−1
αs x
−s
!r−1
+ fN (x)
Close
Quit
Page 9 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
+
n
X
(r − 1)ar xn−2 fN (x) + O(xn−N −3 ).
r=1
Note that O(xn−2 fN (x)) = O(xn−N −2 ). Our assumption for N implies that
coefficients of powers of x down as far as xn−N −1 on the right match the corresponding coefficients in ∆Pa (x, 1), so it follows that
n
X
(r − 1)ar fN (x) = cx−N + O(x−N −1 ),
r=1
where the coefficient c is equal to the difference between the coefficients of
−s r−1
xn−N −2 in ∆Pa (x, 1) and in Σnr=1 ar xn−1 (1 + ΣN
. The coefficient
s=1 αs x )
of fN (x) is nonzero because Ωn−1 (a) is nonzero. Let αN +1 := c/Σnr=1 (r−1)ar .
Then E1 (a) is
!
N
+1
X
−s
y =x 1+
+ O(x−N −1 ).
αs x
s=1
The theorem now follows by induction on N.
Under the conditions of Theorem 3.1, the equipoise curve E1 (a) has an
asymptotic expansion with coefficient sequence α = (α1 , α2 , . . .) as x → ∞.
Since W2 (a) = Σnr=1 (r − 1)ar , the proof of Theorem 3.1 shows that αN =
cN /W2 (a), where cN is the difference between the coefficients of x−N in the
expansions
∞
X
∆Pa (x, 1)
−1
−2
=
Σ
(a)
+
Σ
(a)x
+
Σ
(a)x
+
.
.
.
=
Σk+1 (a)x−k
1
2
3
n−1
x
k=0
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
and
n
X
r=1
ar
1+
N
−1
X
!r−1
αs x
−s
:=
s=1
∞
X
CN,k (a)x−k ,
k=0
so cN = ΣN +1 (a) − CN,N (a). Of course, we have yet to determine the coefficients CN,k (a), but note immediately that CN,k (a) = 0 for all sufficiently large
k.
Let d := (d1 , . . . , dN −1 ) ∈ (Z+ )N −1 be a nonnegative integer sequence such
−1
N −1
that ΣN
s=1 sds = k and Σs=1 ds = m. Then d is a partition of k with length
N − 1 and weight m. Corresponding to each d with weight m ≤ r − 1, there is
r−1
N −1
αs x−s
, with coefficient
a term in x−k in the expansion of 1 + Σs=1
(r − 1)!
d −1
· αd1 αd2 . . . αNN−1
.
(r − m − 1)!d1 !d2 ! . . . dN −1 ! 1 2
For convenience we abbreviate such expressions with the following compact
notation for the product
N
−1
Y
α(d) :=
αsds
s=1
and the multinomial coefficient
Σ(d)
m!
:= N −1 ,
d
Πs=1 ds !
where Σ(d) = m. Thus the coefficient of the term in x−k corresponding to d in
−1
−s r−1
the expansion of 1 + ΣN
becomes
s=1 αs x
r − 1 Σ(d)
α(d),
Σ(d)
d
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
where the first factor is the binomial coefficient r−1
, which by definition is 0
m
when m > r − 1.
Let P (k, N − 1, m) ⊆ (Z+ )N −1 be the set of all partitions of k with length
−1
−s r−1
N − 1 and weight m. Then in the expansion of Σnr=1 ar 1 + ΣN
s=1 αs x
the coefficient of the term in x−k corresponding to any particular d ∈ P (k, N −
1, m) is
n X
r−1
m
m
ar
α(d) =
α(d)Wm+1 (a).
m
d
d
r=1
Summing over all partitions in P (k, N − 1, m) and all relevant weights
m yields CN,k (a), the total coefficient of x−k . When k = N we obtain the
coefficient CN,N (a) needed for αN . Simplifying notation with P (N, m) :=
P (N, N − 1, m), and noting that P (N, 1) = ∅, we have
Theorem 3.2. For any finite sequence a ∈ (R+ )n with n ≥ 2 and Ωn−1 (a)
nonzero, the asymptotic expansion of the equipoise curve E1 (a) is
!
∞
X
y ∼x 1+
αs x−s
as x → ∞,
s=1
where the coefficient sequence α := (α1 , α2 , . . .) ∈ R∞ is given by
αN =
ΣN +1 (a) − CN,N (a)
W2 (a)
for each N ≥ 1, with
CN,N (a) =
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
N
X

X

m=2
d∈P (N,m)

m
α(d) Wm+1 (a).
d
Page 12 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
In particular, when N = 1 we have P (1, m) = ∅ so C1,1 (a) = 0, since its
inner sum is empty. This gives α1 = Σ2 (a)/ W2 (a), consistent with Theorem
2.3. Again, when N = 2 the sequence (2) ∈ R1 is the unique partition
of 2 with
2
length 1, so P (2, 2) = {(2)} and the inner sum for C2,2 (a) is 2 α(2) = α12 ,
whence C2,2 (a) = α12 W3 (a). Then α2 = (Σ3 (a) − α12 W3 (a))/ W2 (a), again
consistent with Theorem 2.3. Substituting here for α1 and suppressing the argument a to simplify notation yields
α2 =
Σ3 W22 − Σ22 W3
.
W23
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
When N = 3 we have P (3, 2) = {(1, 1)} and P (3, 3) = {(3, 0)}, so Theorem
3.2 yields α3 from C3,3 (a) = 2α1 α2 W3 (a) + α13 W4 (a). Substituting for α1 and
α2 and suppressing the argument a now yields
Σ W 4 − Σ32 W2 W4 + 2Σ32 W32 − 2Σ2 Σ3 W22 W3
α3 = 4 2
.
W25
Evidently continuing this process will yield an expression for any αN just
in terms of the elementary symmetric functions and the binomially-weighted
functions of the sequence a. In the next theorem we characterize the summands
in this explicit expression for αN , but first we introduce some notation. For any
integer sequence d ∈ (Z+ )N let
Σ(d) :=
N
Y
r=1
Σr+1 (a)dr
and W(d) :=
N
Y
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Wr+1 (a)dr .
Page 13 of 30
r=1
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
As previously, we shall usually suppress explicit mention of the argument a
from expressions of this type. We also need the following family of partition
pairs:
(
N
X
+ N
rdr = N,
Q(N ) := (d, e) : d, e ∈ (Z ) ,
r=1
N
X
rer = 2N − 2,
r=1
N
X
)
(dr + er ) = 2N − 1 ,
r=1
that is, pairs (d, e) of partitions of N and 2N − 2 respectively, each of length
N, with sum of weights equal to 2N − 1. (This places no effective restriction
on d, but does constrain e significantly.)
Theorem 3.3. For each integer N ≥ 1, the coefficient αN in the asymptotic
expansion of the equipoise curve E1 (a) satisfies an identity of the form
X
αN W22N −1 =
c(d, e)Σ(d)W(e),
(d,e)∈Q(N )
where each coefficient c(d, e) is an integer dependent only on the partition pair
(d, e).
Proof. Note that Q(1) = {((1), (0))} and α1 W2 = Σ2 , so the theorem holds
when N = 1, with c((1), (0)) = 1. Now fix N > 1, and suppose inductively
that the theorem holds for all αs with 1 ≤ s < N. By Theorem 3.2,


N
X
X M 
αN W22N −1 = ΣN +1 W22N −2 −
α(D) W22N −2 WM +1 .
D
M =2
D∈P (N,M )
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
The first term on the right is of the required form, since it is c(d, e)Σ(d)W(e)
with c(d, e) = 1, where d = (0, . . . , 0, 1), e = (2N − 2, 0, . . . , 0) ∈ (Z+ )N are
length N partitions of N and 2N − 2 respectively, with sum of weights 2N − 1.
Now consider the outer sum on the right in the αN identity. Each summand
is of the form


X M 
W22N −M α(D) W2M −2 WM +1 .
D
D∈P (N,M )
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
N −1
(2s − 1)Ds for each D ∈ P (N, M ), we have
Since 2N − M = Σs=1
W22N −M α(D) =
N
−1
Y
s=1
(αs W22s−1 )Ds =
N
−1
Y
Ds

X

s=1
Roger B. Eggleton and
William P. Galvin
c(d, e)Σ(d)W (e)
(d,e)∈Q(s)
where the last step is by hypothesis. If (d, e), (d0 , e0 ) ∈ Q(s) then
Σ(d)W(e) · Σ(d0 )W(e0 ) = Σ(d + d0 )W(e + e0 ),
so it follows that every term in the expansion of the sum over Q(s), raised to
the power Ds , is of the form c(d, e)Σ(d)W(e) where c(d, e) is an integer and
d, e ∈ (Z+ )s are partitions of sDs and (2s − 2)Ds respectively, with sum of
weights (2s − 1)Ds . To calculate the product over s, we modify these length
s partitions by adjoining a further N − s zero terms to each. Partitions corresponding to different values of s can then be added. The sum of N −1 pairs, one
for each value of s, is a pair (d, e) of length N partitions, where d is a partition
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
−1
N −1
of ΣN
s=1 sDs = N and e is a partition of Σs=1 (2s − 2)Ds = 2N − 2M, and the
N −1
sum of weights of d and e is Σs=1
(2s − 1)Ds = 2N − M. Before we sum over
M, recall that each such term is multiplied by W2M −2 WM +1 = W (e∗ ), where
e∗ ∈ (Z+ )N has e∗1 = M − 2, e∗M = 1 and all other terms 0. Thus e∗ is a length
N partition of 2M − 2 with weight M − 1. Hence (d, e + e∗ ) is a pair of length
N partitions of N and 2N − 2 respectively, with sum of weights 2N − 1, so
(d, e + e∗ ) ∈ Q(N ). This does not depend explicitly on M, so the sum over
M is a sum of terms of the form c(d, e)Σ(d)W(e) where (d, e) ∈ Q(N ). All
coefficients c(d, e) involve sums of products of multinomial coefficients and
integer coefficients from αs with 1 ≤ s < N, so every c(d, e) is an integer. The
theorem now follows by induction on N.
For example, the set Q(4) comprises eight partition pairs, each pair being a
length 4 partition of 4 and a length 4 partition of 6, with sum of weights 7. Thus
α4 W27 is a sum of eight products of Σk ’s and Wk ’s, with coefficients as noted:
(d, e) ∈ Q(4)
c(d, e)
((0, 0, 0, 1), (6, 0, 0, 0))
+1
((1, 0, 1, 0), (4, 1, 0, 0))
−2
((0, 2, 0, 0), (4, 1, 0, 0))
−1
((2, 1, 0, 0), (3, 0, 1, 0))
−3
(d, e) ∈ Q(4)
((2, 1, 0, 0), (2, 2, 0, 0))
((4, 0, 0, 0), (2, 0, 0, 1))
((4, 0, 0, 0), (1, 1, 1, 0))
((4, 0, 0, 0), (0, 3, 0, 0))
c(d, e)
+6
−1
+5
−5
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
In particular, the term in α4 W27 with the largest coefficient is 6Σ22 Σ3 W22 W32 , and
the term independent of W2 is −5Σ42 W33 .
Corollary 3.4. For any positive integer N, there are integers c(d, e) corresponding to pairs of partitions (d, e) ∈ Q(N ) such that the degree 2N − 1
Quit
Page 16 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
homogeneous polynomial in 2N variables,
FN (u1 , . . . , uN ; v1 , . . . , vN ) :=
X
(d,e)∈Q(N )
c(d, e)
N
Y
r=1
udr r
N
Y
vses ,
s=1
when evaluated at ur = Σr+1 (a), vs = Ws+1 (a), 1 ≤ r, s ≤ N, takes the value
FN (Σ2 (a), . . . , ΣN +1 (a); W2 (a), . . . , WN +1 (a)) = αN W2 (a)2N −1 .
From the 2N variables ur , vs (1 ≤ r, s ≤ N ) let us form 2N − 1 “rational”
variables:
vs+1
ur
(1 ≤ r ≤ N ) and τs :=
(1 ≤ s ≤ N − 1).
ρr :=
vr
v1
Then F1 (u1 ; v1 ) = u1 = ρ1 v1 and F2 (u1 , u2 ; v1 , v2 ) = u2 v12 −u21 v2 = (ρ2 −ρ21 )v12 v2 ,
whence
F1 (u1 ; v1 )
= ρ1
v1
and
F2 (u1 , u2 ; v1 , v2 )
= (ρ2 − ρ21 )τ1 .
3
v1
A corresponding identity can be obtained for each FN . Indeed if N ≥ 2 then


N
−1
X
X m 
cN,N = α1N WN +1 +
α(d) Wm+1 ,
d
m=2
d∈P (N,m)
by Theorem 3.2, whence
N −2
FN (u1 , . . . , uN ; v1 , . . . , vN ) = uN v12N −2 − uN
vN + RN ,
1 v1
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
where RN = RN (u1 , . . . , uN −1 ; v1 , . . . , vN −1 ) is a polynomial which does not
involve the variables uN and vN . Now
N −2
2N −1
uN v12N −2 − uN
vN = (ρN − ρN
,
1 v1
1 )τN −1 v1
whence induction on N utilizing Theorem 3.2 and Corollary 3.4 establishes the
general identity for FN in Corollary 3.5 below. Once again, some additional
notation allows us to express the result compactly. For any d ∈ (Z+ )N and
e ∈ (Z+ )N −1 we write
ρ(d) :=
N
Y
ρdr r
and τ (e) :=
r=1
N
−1
Y
τses .
s=1
Also for any partition pair (d, e) ∈ Q(N ) note that the sequence d∗ ∈ (Z+ )N −1
given by d∗ := ΩN −1 (d + e) is a length N − 1 partition of N − 1, since
N
−1
X
rd∗r
=
r=1
=
N
X
r=2
N
X
r=1
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
(r − 1)(dr + er )
rdr +
N
X
r=1
rer −
N
X
JJ
J
(dr + er )
II
I
Go Back
r=1
= N + (2N − 2) − (2N − 1) = N − 1.
Let P (N ) be the set of all length N partitions of N, and for each d∗ ∈ P (N −1),
let us define the subfamily of partition pairs Q∗ (d∗ ) := {(d, e) ∈ Q(N ) :
ΩN −1 (d + e) = d∗ }. Then induction establishes
Close
Quit
Page 18 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
Corollary 3.5. For any integer N ≥ 2, the polynomial FN satisfies the identity
X
FN (u1 , . . . , uN ; v1 , . . . , vN )
=
c(d, e)ρ(d)τ (ΩN −1 (d + e))
2N −1
v1
(d,e)∈Q(N )
X
fd∗ (ρ1 , . . . , ρN )τ (d∗ ),
=
d∗ ∈P (N −1)
where
X
fd∗ (ρ1 , . . . , ρN ) :=
c(d, e)ρ(d).
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
(d,e)∈Q∗ (d∗ )
In particular we have
f(1) = ρ2 − ρ21 ,
f(2,0) = 2ρ31 − 2ρ1 ρ2 ,
f(3,0,0) = 6ρ21 ρ2 −ρ22 −5ρ41 ,
Roger B. Eggleton and
William P. Galvin
f(0,1) = ρ3 − ρ31 ,
f(1,1,0) = 5ρ41 −3ρ21 ρ2 −2ρ1 ρ3 ,
f(0,0,1) = ρ4 −ρ41 .
For each d∗ ∈ P (N − 1), the polynomial fd∗ in the N variables ρ1 , . . . , ρN
has coefficients which are a subfamily of the coefficients introduced in Theorem
3.3. Each (d, e) ∈ Q(N ) determines a unique d∗ = ΩN −1 (d + e) ∈ P (N − 1),
so the families {c(d, e) : (d, e) ∈ Q∗ (d∗ )} comprise a partition of the family
of coefficients {c(d, e) : (d, e) ∈ Q(N )} introduced in Theorem 3.3. For each
−1 ∗
d∗ ∈ P (N − 1), let e ∈ (Z+ )N be such that ΩN −1 (e) = d∗ and e1 = ΣN
r=1 dr .
Then e is a length N partition of 2N − 2 with weight 2N − 1, and (d, e) ∈
Q∗ (d∗ ) when d = (N, 0, . . . , 0). Hence fd∗ contains the term c(d, e)ρN
1 , and it
follows that fd∗ is of degree N. In particular our earlier calculations show that
∗
fd∗ (ρ1 , . . . , ρN ) = ρN − ρN
1 when d = (0, . . . , 0, 1) ∈ P (N − 1), and when
evaluated at ρ1 = . . . = ρN = 1 this polynomial takes the value 0, so the sum
of its coefficients is 0. Then induction establishes
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
Corollary 3.6. For any N ≥ 2 and each d∗ ∈ P (N − 1), the polynomial
fd∗ (ρ1 , . . . , ρN ) is of degree N, and the sum of its coefficients is 0.
Since each fd∗ satisfies fd∗ (1, . . . , 1) = 0, it immediately follows that we
have
Corollary 3.7. For any integer N ≥ 2, the polynomial FN (u1 , . . . , uN ; v1 , . . . , vN )
has sum of coefficients equal to 0, and in fact satisfies the identity FN (u1 , . . . , uN ;
u1 , . . . , uN ) = 0.
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
4.
Particular Evaluations of the Asymptotic
Expansion
We shall now consider evaluations of the coefficient sequence α = (α1 , α2 , . . .)
of the asymptotic expansion of the equipoise curve E1 (a), for particular choices
of a ∈ (R+ )n .
First, note that ΣN +1 (a) = 0 = WN +1 (a) for all N ≥ n. This causes no
concern if we use Corollary 3.4 to determine αN by evaluating FN at ur =
Σr+1 (a), vs = Ws+1 (a), 1 ≤ r, s ≤ N. On the other hand, it is not immediately obvious how we should use Corollary 3.5 to determine αN when N ≥ n,
since the rational variable ρr = ur /vr does not have a stand-alone value when
ur = 0, vr = 0. However, for each partition pair (d, e) ∈ Q(N ) the proddr +er
uct ρ(d)τ (ΩN −1 (d + e)) contains the factor ρdr r τr−1
= udr r vrer /v1dr +er , which
takes the value 0 when ur = 0 and vr = 0, since Ωn−1 (a) nonzero ensures
that v1 = W2 (a) is nonzero. Thus, Corollary 3.5 yields αN as the value of
FN /v12N −1 when ur = Σr+1 (a), vs = Ws+1 (a), 1 ≤ r, s ≤ N, by noting
that the only products fd∗ τ (d∗ ) that can be nonzero correspond to partitions
d∗ ∈ P (N − 1) with ΩN −n (d∗ ) = 0.
Example 4.1. If a ∈ (R+ )2 with Ω1 (a) nonzero, it is trivial to verify that the
equipoise curve E1 (a) is the straight line y = x + α1 , with α1 = a1 and αN = 0
for N ≥ 2.
+ 3
Example 4.2. If a ∈ (R ) with Ω2 (a) nonzero, the equipoise curve E1 (a)
is a hyperbola with asymptote y = x + α1 . Corollary 3.5 yields αN as the
value of fd∗ τ (d∗ ) for d∗ = (N − 1, 0, . . . , 0) ∈ P (N − 1), with the evaluation
ρ1 = Σ2 (a)/ W2 (a), ρ2 = Σ3 (a)/ W3 (a) and τ1 = W3 (a)/ W2 (a), noting that
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 21 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
any products of ρ2 and τ1 are equal to zero if a3 = 0. We have
α1 = ρ1 , α2 = (ρ2 −ρ21 )τ1 , α3 = (2ρ31 −2ρ1 ρ2 )τ12 , α4 = (6ρ21 ρ2 −ρ22 −5ρ41 )τ13 ,
and so on. However we do not explicitly know the coefficients of f(N −1,0,...,0)
in general. On the other hand, Theorem 3.2 conveniently determines αN recursively in this case. In fact, with α1 and α2 as determined above, we have all
later terms given by the recurrence
!
N −1
W3 (a) X
αN = −
αs αN −s
for N ≥ 3.
W2 (a) s=1
The substitution βN := −τ1 αN for N ≥ 1 converts the recurrence to a pure
convolution
N
−1
X
βN =
βs βN −s
for N ≥ 3,
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
s=1
corresponding to the classical recurrence satisfied by Catalan numbers, but now
with initial conditions β1 = −ρ1 τ1 and β2 = (ρ21 − ρ2 )τ12 . There is an extensive
literature on Catalan numbers. An accessible and readily available discussion
is the subject of Chapter 7 of [3].
JJ
J
II
I
Go Back
Close
Introducing the generating function
2
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
N
F (z) := β1 z + β2 z + . . . + βN z + . . . =
∞
X
Quit
r
βr z ,
Page 22 of 30
r=1
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
and letting Z := τ1 z, we find that F satisfies F (z)2 − F (z) − Z(ρ1 + ρ2 Z) = 0,
so
p
1 − 1 + 4Z(ρ1 + ρ2 Z)
F (z) =
.
2
Binomial series expansion now leads to an explicit closed-form solution for βN ,
whence
bN/2c
X
(−1)s
2N − 2s − 1
N −1
−2s s
αN = (−τ1 )
ρN
ρ2 for N ≥ 1.
1
2N
−
2s
−
1
N
−
2s,
s,
N
−
s
−
1
s=0
Here the quotient of the trinomial coefficient with 2N − 2s − 1 can be regarded
as a generalized Catalan number. For example, this explicit solution readily
yields
9
1
7
1
5
4 1
5
3
2
ρ −
ρ ρ2 +
ρ1 ρ2 ,
α5 = τ1
9 5, 0, 4 1 7 3, 1, 3 1
5 1, 2, 2
whence α5 = (14ρ51 −20ρ31 ρ2 +6ρ1 ρ22 )τ14 . Note by Corollary 3.6 that each fd∗ has
coefficient sum zero, so the generalised Catalan numbers have zero alternating
sum for N ≥ 2 :
bN/2c
X
2N − 2s − 1
(−1)s
= 0.
2N − 2s − 1 N − 2s, s, N − s − 1
s=0
Alternatively, this identity can be deduced from the quadratic identity for F (z)
when ρ21 = ρ2 . Putting Z ∗ := ρ1 τ1 z then gives F (z)2 − F (z) − Z ∗ (1 + Z ∗ ) = 0,
and binomial series expansion of the solution yields the zero alternating sum
noted.
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
Example 4.3. Let us now consider the constant sequence
a = 1 ∈ (R+ )n in
n
which each term is equal to 1. Then Σk (1) = k = Wk (1) for 1 ≤ k ≤ n,
so in this case α1 = 1 and Corollaries 3.4 and 3.7 imply that αN = 0 for
N ≥ 2. Hence, as in Example 4.1, the equipoise curve E1 (1) is the straight line
y = x + 1. This is confirmed by noting that ∆P1 (x, 1) = (x + 1)n − xn and
S1 (x, y) = (xn − y n )/(x − y), so ∆P1 (x, 1) = S1 (x, y) holds when y = x + 1.
Example 4.4. Let δ := (δ1 , δ2 , . . . , δn ) ∈ Rn be a sequence in which every term
satisfies |δr | ≤ for some small strictly positive ∈ R+ . Then a := 1 + δ ∈
(R+ )n is a small perturbation of the constant sequence 1. Let Σ0 (δ) := 1. Then
for each k ≥ 1 we have
Σk (1 + δ) =
k X
n−s
s=0
k−s
Σs (δ)
and
n
Wk (1 + δ) =
+ Wk (δ).
k
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
It is convenient
Wk by dividing by Σk (1) =
to scale the functions Σk and
n
+ n
Wk (1) = k when 1 ≤ k ≤ n : for all a ∈ (R ) we define
Σ∗k (a) := Σk (a)/ Σk (1)
and
Wk∗ (a) := Wk (a)/Wk (1).
(It can easily be shown that Σ∗k (a) is the expected value of the product of terms
in a k-term subsequence of a, and Wk∗ (a) is the expected value of the last term
in a k-term subsequence of a.) It is also appropriate to define Σ∗0 (δ) = 1. Then
for 1 ≤ k ≤ n the earlier identities become
k X
k ∗
∗
Σk (1 + δ) =
Σ (δ)
s s
s=0
and
Wk∗ (1
+ δ) = 1 +
Wk∗ (δ).
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
We keep n fixed and let → 0+ , so O(Σ∗k (δ)) = O(k ) and O(Wk∗ (δ)) = O().
In particular,
α1 = 1 + 2Σ∗1 (δ) − W2∗ (δ) + O(2 ).
For any integer s ≥ 0, put
n
n−2
s+2
n
=
λs :=
2
s
2
s+2
N −1 ds
and for any d ∈ (Z+ )N −1 define λ(d) := Πs=1
λs . Now evaluating the coefficient αN at 1 + δ using Corollaries 3.4 and 3.5 is convenient so long as we
know fd∗ explicitly for each d∗ ∈ P (N − 1). We have O(fd∗ ) = O() because
ρr = 1 + O() for 1 ≤ r ≤ n − 1 and fd∗ has zero coefficient sum by Corollary
3.6. As τs = λs + O() for 1 ≤ s ≤ n − 2, it follows for N ≥ 2 that
X
αN =
fd∗ (ρ1 , . . . , ρN )λ(d∗ ) + O(2 )
fd∗ (ρ1 , . . . , ρN )
(d,e)∈Q∗ (d∗ )
Title Page
JJ
J
where
=
Roger B. Eggleton and
William P. Galvin
Contents
d∗ ∈P (N −1)
X
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
c(d, e)
N Σ∗1 (δ)
+
N
X
!
∗
∗
(δ) + O(2 ).
dr Σ1 (δ) − Wr+1
r=1
On the other hand, using Theorem 3.2 to evaluate αN at 1 + δ yields α1 as
II
I
Go Back
Close
Quit
Page 25 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
above and for N ≥ 2 yields αN via the recurrence
αN = −
N
−2
X
(s + 1)λs αN −s − N λN −1 (Σ∗1 (δ) − W2∗ (δ))
s=1
+ λN −1 Σ∗1 (δ) − WN∗ +1 (δ) + O(2 ).
It follows by induction for N ≥ 2 that αN has zero sum for the coefficients of
the family of functions Σ∗1 (δ) and Wk∗ (δ) with 2 ≤ k ≤ N + 1.
Note in particular the special case in which An−1 (δ) = 0, so δr = 0 for
1 ≤ r ≤ n − 1 and |δn | ≤ . Then Σ∗1 (δ) = δn /n and Wk∗ (δ) = kδn /n for
2 ≤ k ≤ n, so α1 = 1 and αN = 0 for N ≥ 2. This is confirmed directly by
checking that ∆P1+δ (x, 1) = (x + 1)n − xn + δn (x + 1)n−1 and S1+δ (x, y) =
(xn − y n )/(x − y) + δn y n−1 are equal precisely when y = x + 1.
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
5.
Selected Inequalities
To conclude, let us briefly sample some of the inequalities between the polynomials ∆Pa (x, 1) and Sa (x, y) which are consequences of the preceding asymptotic analysis.
Case 5.1. a = 1 ∈ (R+ )n with n ≥ 2.
In this case the equipoise curve is y = x + 1, and simple but elegant inequalities
are already implied by Theorems 2.2 and 2.3. For instance, ∆P1 (1, 1) = 2n − 1
and S1 (1, 3) = (3n − 1)/2. The point (1, 3) lies above the equipoise curve, so
is in the S-region, and Theorem 2.2 implies
3n − 1
2n − 1 <
.
2
Indeed, the lines y = x + 2 and y = x + 12 lie, respectively, in the S-region and
the ∆P -region, so for x > 0 we have
n
1
(x + 1)n − xn
(x + 2)n − xn
n
−x <
<
.
x+
2
2
4
These inequalities would usually be deduced from the convexity of y = xn , and
actually hold for all x ∈ R when n is even.
Since α1 = 1 and αN +1 = 0 for N ≥ 1 when a = 1, less familiar inequalities can be derived by noting that the curves y = x+1+x−N and y = x+1−x−N
lie, respectively, in the S-region and the ∆P -region when N ≥ 1. Thus for
x > 0 we have
n
n
x + 1 − x1N − xn
x + 1 + x1N − xn
n
n
< (x + 1) − x <
1 − x1N
1 + x1N
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
Once again these inequalities could be deduced from convexity of y = xn , but
now their form is more naturally suggested by the asymptotic expansion of the
equipoise curve.
Case 5.2. a = 1 + δ ∈ (R+ )n with n ≥ 3, and there is some small strictly
positive ∈ R+ such that |δr | ≤ for 1 ≤ r ≤ n.
Let us consider the special case in which δ1 = −, δn = and δr = 0 for
2 ≤ r ≤ n − 1. Then
∆P1+δ (x, 1) = (x + 1)n − xn − 2 (x + 1)n−2
y n − xn
+ (y n−1 − xn−1 ).
S1+δ (x, y) =
y−x
and
In this case Σ∗1 (δ) = 0, Σ∗2 (δ) = −2/n(n − 1), Σ∗k (δ) = 0 for 3 ≤ k ≤ n, and
Wk∗ (δ) = k/n for 2 ≤ k ≤ n. From Example 4.4 we have
2
α1 = 1− +O(2 ),
n
n−2
α2 =
+O(2 ),
3n
(n + 1)(n − 2)
α3 = −
+O(2 ).
18n
Then α2 > 0 and α3 < 0, so the curves y = x + α1 and y = x + α1 + α2 x−1
lie, respectively, below and above the equipoise curve for sufficiently large x.
Hence
(x + α1 )n − xn
+ (x + α1 )n−1 − xn−1
α1
< (x + 1)n − xn − 2 (x + 1)n−2
n
x + α1 + αx2 − xn
α2 n−1
n−1
<
+ x + α1 +
−x
,
α1 + αx2
x
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 28 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
where α1 and α2 take the exact values
n
− 2
2
and α2 =
α1 = n
+ (n − 1)
2
n
3
− (n − 2)2 − α12 n3 +
n
+ (n − 1)
2
n−1
2
.
Case 5.3. a := (a, b, c) ∈ (R+ )3 with Ω2 (a) = (b, c) 6= (0, 0).
As shown in Example 4.2, for N ≥ 1 each αN is a function of ρ1 , ρ2 and τ1 in
this case. If c = 0, we easily verify that α1 = a and αN = 0 for N ≥ 2. Now
suppose c > 0. Then ρ1 , ρ2 and τ1 have stand-alone values, and α2 = 0 precisely
when ρ2 = ρ21 . But then αN = 0 for N ≥ 2, by the previously noted alternating
sum identity for the generalised Catalan numbers. If ρ2 > ρ21 then α2 > 0 and
the summation identity for αN in Example 4.2 implies (−1)N αN > 0, so αN
alternates in sign for N ≥ 2. Similarly if ρ2 < ρ21 then α2 < 0 and αN alternates
in sign for N ≥ 2. As in Case 5.2 above we can deduce relevant inequalities. In
particular, if ρ2 > ρ21 then for sufficiently large x we have
2
2
ax + bx(x + α1 ) + c(x + α1 )
< (x + a)(x + b)(x + c) − x3
α2 α2 2
< ax2 + bx x + α1 +
+ c x + α1 +
,
x
x
where α1 and α2 have their exact values, given explicitly in Example 4.2.
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 29 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au
References
[1] T.J. BROMWICH, An Introduction to the Theory of Infinite Series, Chap. 6
(MacMillan, 1965).
[2] R.B. EGGLETON and W.P. GALVIN, A polynomial inequality generalising an integer inequality, J. Inequal. Pure and Appl. Math., 3 (2002), Art.
52, 9 pp.
[ONLINE: http://jipam.vu.edu.au/v3n4/043_02.html]
[3] P. HILTON, D. HOLTON and J. PEDERSEN, Mathematical Vistas – From
a Room with Many Windows, Springer (2002).
[4] K. WEIERSTRASS, Über die Theorie der analytischen Facultäten, Crelle’s
Journal, 51 (1856), 1–60.
Asymptotic Expansion of the
Equipoise Curve of a
Polynomial Inequality
Roger B. Eggleton and
William P. Galvin
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 30 of 30
J. Ineq. Pure and Appl. Math. 3(5) Art. 84, 2002
http://jipam.vu.edu.au