Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 5, Article 84, 2002
ASYMPTOTIC EXPANSION OF THE EQUIPOISE CURVE OF A POLYNOMIAL
INEQUALITY
ROGER B. EGGLETON AND WILLIAM P. GALVIN
D EPARTMENT OF M ATHEMATICS
I LLINOIS S TATE U NIVERSITY
N ORMAL , IL 61790-4520, USA.
roger@math.ilstu.edu
S CHOOL OF M ATHEMATICAL AND P HYSICAL S CIENCES
U NIVERSITY OF N EWCASTLE
C ALLAGHAN , NSW 2308, AUSTRALIA .
mmwpg@cc.newcastle.edu.au
Received 28 August, 2002; accepted 8 November, 2002
Communicated by H. Gauchman
A BSTRACT. 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 y n−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.
Key words and phrases: Polynomial inequality, Catalan numbers.
2000 Mathematics Subject Classification. 26D05, 26C99, 05A99.
1. I NTRODUCTION
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
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
The first author gratefully acknowledges the hospitality of the School of Mathematical and Physical Sciences, University of Newcastle,
throughout the writing of this paper.
091-02
2
ROGER B. E GGLETON AND W ILLIAM P. G ALVIN
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
∆Pa (x, t) := Pa (x, t) − Pa (x, 0) = Pa (x, t) − xn
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
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 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)}.
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.
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
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A SYMPTOTIC E XPANSION OF THE E QUIPOISE C URVE OF A P OLYNOMIAL I NEQUALITY
3
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.
2. W IDER R ANGE
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
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
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
x + min An−1 (a) ≤ y0 (x) ≤ x + max An−1 (a).
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
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.
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
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4
ROGER B. E GGLETON AND W ILLIAM P. G ALVIN
The kth binomially-weighted sum Wk of a ∈ Rn is the sequence function
n X
r−1
ar .
Wk (a) :=
k−1
r=1
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 → ∞,
where
α := Σ2 (a)/W2 (a)
β := (Σ3 (a) − α2 W3 (a))/W2 (a).
and
Note that β = 0 if n = 2. We will extend Theorem 2.3 in the next section.
3. A SYMPTOTIC E XPANSION OF THE E QUIPOISE C URVE
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
αs x
as x → ∞.
y ∼x 1+
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 ) as x → ∞. Then
∆Pa (x, 1) = Sa (x, y)
=
n
X
ar x
n−r
x 1+
r=1
=
n
X
N
X
!r−1
!
αs x
−s
+ fN (x)
s=1
ar xn−1
1+
N
X
s=1
r=1
!r−1
αs x−s
+
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
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
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A SYMPTOTIC E XPANSION OF THE E QUIPOISE C URVE OF A P OLYNOMIAL I NEQUALITY
5
where the coefficient c is equal to the difference between the coefficients of xn−N −2 in ∆Pa (x, 1)
−s r−1
and in Σnr=1 ar xn−1 (1 + ΣN
. The coefficient of fN (x) is nonzero because Ωn−1 (a)
s=1 αs x )
n
is nonzero. Let αN +1 := c/Σr=1 (r − 1)ar . Then E1 (a) is
!
N
+1
X
y =x 1+
αs x−s + O(x−N −1 ).
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
xn−1
k=0
and
n
X
ar
1+
r=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.
−1
Let d := (d1 , . . . , dN −1 ) ∈ (Z+ )N −1 be a nonnegative integer sequence such that ΣN
s=1 sds =
−1
k and ΣN
to
s=1 ds = m. Then d is a partition of k with length N −1 and weight m. Corresponding
N −1
−k
−s r−1
,
each d with weight m ≤ r − 1, there is a term in x in the expansion of 1 + Σs=1 αs x
with coefficient
(r − 1)!
d −1
· α1d1 α2d2 . . . αNN−1
.
(r − m − 1)!d1 !d2 ! . . . dN −1 !
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 the expansion
r−1
N −1
of 1 + Σs=1
αs x−s
becomes
r − 1 Σ(d)
α(d),
Σ(d)
d
where the first factor is the binomial coefficient r−1
, which by definition is 0 when m > r − 1.
m
+ N −1
Let P (k, N − 1, m) ⊆ (Z )
be the set of all partitions of k with length N − 1 and weight
−1
n
−s r−1
m. Then in the expansion of Σr=1 ar 1 + ΣN
the coefficient of the term in x−k
s=1 αs x
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
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
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6
ROGER B. E GGLETON AND W ILLIAM P. G ALVIN
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
as x → ∞,
y ∼x 1+
αs x−s
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) =
N
X

X

m=2
d∈P (N,m)

m
α(d) Wm+1 (a).
d
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 length 1, so P (2, 2) = {(2)} and the inner sum for
C2,2 (a) is 22 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
Σ W 2 − Σ2 W
α2 = 3 2 3 2 3 .
W2
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
Σ4 W24 − Σ32 W2 W4 + 2Σ32 W32 − 2Σ2 Σ3 W22 W3
.
α3 =
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
Σr+1 (a)dr
and W(d) :=
r=1
N
Y
Wr+1 (a)dr .
r=1
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
Q(N ) := (d, e) : d, e ∈ (Z ) ,
rdr = N,
r=1
N
X
r=1
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
rer = 2N − 2,
N
X
)
(dr + er ) = 2N − 1 ,
r=1
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A SYMPTOTIC E XPANSION OF THE E QUIPOISE C URVE OF A P OLYNOMIAL I NEQUALITY
7
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 )
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 )
−1
Since 2N − M = ΣN
s=1 (2s − 1)Ds for each D ∈ P (N, M ), we have

Ds
N
−1
N
−1
Y
X
Y

(αs W22s−1 )Ds =
c(d, e)Σ(d)W (e)
W22N −M α(D) =
s=1
s=1
(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
−1
each value of s, is a pair (d, e) of length N partitions, where d is a partition of ΣN
s=1 sDs = N
N −1
and e is a partition of Σs=1 (2s − 2)Ds = 2N − 2M, and the sum of weights of d and e is
−1
ΣN
s=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.
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
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8
ROGER B. E GGLETON AND W ILLIAM P. G ALVIN
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
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 homogeneous polynomial in 2N
variables,
N
N
X
Y
Y
dr
ur
vses ,
FN (u1 , . . . , uN ; v1 , . . . , vN ) :=
c(d, e)
r=1
(d,e)∈Q(N )
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:
ur
vs+1
ρr :=
(1 ≤ r ≤ N ) and τs :=
(1 ≤ s ≤ N − 1).
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 .
v13
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
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
r=1
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
ρdr r
and τ (e) :=
N
−1
Y
τses .
s=1
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A SYMPTOTIC E XPANSION OF THE E QUIPOISE C URVE OF A P OLYNOMIAL I NEQUALITY
9
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 − 1)(dr + er )
r=2
=
N
X
r=1
rdr +
N
X
rer −
r=1
N
X
(dr + er )
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
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))
v12N −1
(d,e)∈Q(N )
X
=
fd∗ (ρ1 , . . . , ρN )τ (d∗ ),
d∗ ∈P (N −1)
where
X
fd∗ (ρ1 , . . . , ρN ) :=
c(d, e)ρ(d).
(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 ,
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 d∗ ∈ P (N − 1), let e ∈ (Z+ )N be such that ΩN −1 (e) = d∗ and
−1 ∗
∗
∗
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
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.
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10
ROGER B. E GGLETON AND W ILLIAM P. G ALVIN
4. PARTICULAR E VALUATIONS OF THE A SYMPTOTIC E XPANSION
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 standalone value when ur = 0, vr = 0. However, for each partition pair (d, e) ∈ Q(N ) the product
dr +er
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.
Example 4.2. If a ∈ (R+ )3 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 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
X
W3 (a)
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
βN =
N
−1
X
βs βN −s
for N ≥ 3,
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].
Introducing the generating function
∞
X
2
N
F (z) := β1 z + β2 z + . . . + βN z + . . . =
βr z r ,
r=1
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
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A SYMPTOTIC E XPANSION OF THE E QUIPOISE C URVE OF A P OLYNOMIAL I NEQUALITY
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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
α5 = τ1
ρ −
ρ ρ2 +
ρ1 ρ2 ,
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
(−1)s
2N − 2s − 1
= 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.
Example 4.3. Let us now consider
the constant sequence a = 1 ∈ (R+ )n in which each term is
n
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
k X
n−s
n
Σk (1 + δ) =
Σs (δ)
and
Wk (1 + δ) =
+ Wk (δ).
k−s
k
s=0
It is convenient to scale the functions Σk and Wk by dividing by Σk (1) = Wk (1) = nk when
1 ≤ k ≤ n : for all a ∈ (R+ )n 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) =
Σs (δ)
and
Wk∗ (1 + δ) = 1 + Wk∗ (δ).
s
s=0
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
λs :=
n
n
n−2
s+2
=
s+2
2
s
2
−1 ds
and for any d ∈ (Z+ )N −1 define λ(d) := ΠN
s=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 )
d∗ ∈P (N −1)
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
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12
ROGER B. E GGLETON AND W ILLIAM P. G ALVIN
where
fd∗ (ρ1 , . . . , ρN ) =
X
c(d, e) N Σ∗1 (δ) +
(d,e)∈Q∗ (d∗ )
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 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)) + λN −1 Σ∗1 (δ) − WN∗ +1 (δ) + O(2 ).
s=1
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.
5. S ELECTED I NEQUALITIES
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 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
2n − 1 <
3n − 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
x+
− xn <
<
.
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
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 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
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
and S1+δ (x, y) =
y n − xn
+ (y n−1 − xn−1 ).
y−x
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A SYMPTOTIC E XPANSION OF THE E QUIPOISE C URVE OF A P OLYNOMIAL I NEQUALITY
13
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
n−2
(n + 1)(n − 2)
α1 = 1 − + O(2 ), α2 =
+ O(2 ), α3 = −
+ O(2 ).
n
3n
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
where α1 and α2 take the exact values
n
− 2
2
α1 = n
and α2 =
+
(n
−
1)
2
n
3
− (n − 2)2 − α12 n3 +
n
+ (n − 1)
2
n−1
2
.
Case 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 2 above we can deduce relevant inequalities. In particular, if ρ2 > ρ21 then
for sufficiently large x we have
ax2 + bx(x + α1 ) + c(x + α1 )2 < (x + a)(x + b)(x + c) − x3
α2 α2 2
+ c x + α1 +
,
< ax2 + bx x + α1 +
x
x
where α1 and α2 have their exact values, given explicitly in Example 4.2.
R EFERENCES
[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.
J. Inequal. Pure and Appl. Math., 3(5) Art. 84, 2002
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