Journal of Inequalities in Pure and Applied Mathematics

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1 Journal of Inequalities in Pure and Applied Mathematics ASYMPTOTIC EXPANSION OF THE EQUIPOISE CURVE OF A POLYNOMIAL INEQUALITY ROGER B. EGGLETON AND WILLIAM P. GALVIN Department of Mathematics Illinois State University Normal, IL , USA. roger@math.ilstu.edu School of Mathematical and Physical Sciences University of Newcastle Callaghan, NSW 2308 Australia. mmwpg@cc.newcastle.edu.au volume 3, issue 5, article 84, Received 28 August, 2002; accepted 8 November, Communicated by: H. Gauchman Abstract Home Page c 2000 Victoria University ISSN (electronic):

2 Abstract For any a := (a 1, a 2,..., a n ) (R + ) n, define P a (x, t) := (x + a 1 t)(x + a 2 t) (x + a n t) x n and S a (x, y) := a 1 x n 1 + a 2 x n 2 y + + a n y n 1. The two homogeneous polynomials P a (x, t) and ts a (x, y) are comparable in the positive octant x, y, t R +. Recently the authors [2] studied the inequality P a (x, t) > ts a (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 P a (x, 1) = S a (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 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. 1 Introduction Wider Range of Validity Equipoise Curve Particular Evaluations of the Asymptotic Expansion Selected Inequalities References Page 2 of 30

3 1. Introduction With any finite real sequence a := (a 1, a 2,..., a n ) R n we associate two homogeneous polynomials, the product polynomial P a (x, t) := (x + a 1 t)(x + a 2 t) (x + a n t) = and the sum polynomial S a (x, y) := a 1 x n 1 + a 2 x n 2 y + + a n y n 1 = n (x + a r t), n a r x n r y r 1. As shown in [2], the first difference of the product polynomial P a (x, t) := P a (x, t) P a (x, 0) = P a (x, t) x n 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 Π n (1 + a r ) and the sum Σ n a r. Indeed Weierstrass [4] derived inequalities equivalent to 1 + n a r < n (1 + a r ) < 1 n (1 a r) < 1 1 n a r when a (R + ) n and 0 < Σ n a r < 1, whence the products Π n (1 + a r ) and Π n (1 a r ) both converge as n if Σ n a r converges to a limit strictly Page 3 of 30

4 less than 1. The first and third of these inequalities correspond to S a (1, 1) < P a (1, 1) and S a (1, 1) < P a (1, 1) respectively, while the middle inequality simply follows from 1 a 2 r 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 a r > 0 for 1 r n. Then the strict inequality P a (x, t) > ts a (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{a r : 1 r n 1}, and the reverse inequality P a (x, t) < ts a (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{a r : 1 r n 1}. The boundary between the P -region and the S-region is the equipoise surface E 2 (a) := {(x, y, t) (R + ) 3 : P a (x, t) = ts a (x, y)}. The polynomials are homogeneous in a, so for any real t we have P ta (x, 1) = P a (x, t) and S ta (x, y) = ts a (x, y), where ta := (ta 1, ta 2,..., ta n ) R n. 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 E 1 (a) := {(x, y) (R + ) 2 : P a (x, 1) = S a (x, y)}. Page 4 of 30

5 This separates the P -region of the positive quadrant x, y R +, where P a (x, 1) > S a (x, y), from the S-region, where P a (x, 1) < S a (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. Page 5 of 30

6 2. Wider Range of Validity To extend the results of [2] it is convenient to introduce some notation. For any sequence a := (a 1, a 2,..., a n ) R n and any integer k in the interval 0 k n, let A k (a) := (a 1, a 2,..., a k ) and Ω k (a) := (a n k+1,..., a n 1, a n ) be, respectively, the initial and final k-term subsequences of a. Thus A n (a) = Ω n (a) = a and, if ω is the empty sequence, A 0 (a) = Ω 0 (a) = ω. Also m(a) := min{a r : 1 r n 1} = min A n 1 (a) and M(a) := max{a r : 1 r n 1} = max A n 1 (a). As in [2] we also use Σ(a) := Σ n a r. First, a simple reformulation of Corollary 2.2 of [2] becomes Theorem 2.1. For any finite sequence a (R + ) n with n 3, and for all strictly positive x, y, t R +, if A n 1 (a) is not constant then for y x+t max A n 1 (a) we have 0 < tσ(a)x n 1 < P a (x, t) < ts a (x, y), while for y x + t min A n 1 (a) and z := min{x, y} we have P a (x, t) > ts a (x, y) tσ(a)z n 1 > 0. To investigate the equality P a (x, 1) = S a (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 y S a(x, y) > 0 Page 6 of 30

7 holds if and only if Ω n 1 (a) 0, where 0 R n 1 is the constant sequence with every term equal to 0. We shall abbreviate this condition by saying if and only if Ω n 1 (a) is nonzero. Then continuity of S a (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 Ω n 1 (a) is nonzero then there is a function y 0 (x) such that < S a (x, y) if y > y 0 (x), P a (x, 1) = S a (x, y) if y = y 0 (x), > S a (x, y) if y < y 0 (x). Furthermore x + min A n 1 (a) y 0 (x) x + max A n 1 (a). As in [2], it is convenient now to define two families of sequence functions Σ k, W k : R n 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 = y 0 (x), the equipoise curve for a. The kth elementary symmetric function Σ k of a R n is the sum of all products Πx as x runs through the k-term subsequences x a, thus Σ k (a) := Σ {Πx : x a, x = k}. In particular, Σ 1 (a) = Σ n a r and Σ 2 (a) = Σ n 1 Σ n s=r+1a r a s if n 2. We extend the definition by setting Σ k (a) = 0 for any integer k > n. Page 7 of 30

8 The kth binomially-weighted sum W k of a R n is the sequence function W k (a) := n ( ) r 1 a r. k 1 In particular, W 1 (a) = Σ n a r and W 2 (a) = Σ n (r 1)a r if n 2. Note that W 1 (a) = Σ 1 (a) holds for any a. Once again we extend the definition by setting W k (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 Ω n 1 (a) is nonzero then the equality P a (x, 1) = S a (x, y) holds for large x when where y = x + α + βx 1 + O(x 2 ) as x, α := Σ 2 (a)/w 2 (a) and β := (Σ 3 (a) α 2 W 3 (a))/w 2 (a). Note that β = 0 if n = 2. We will extend Theorem 2.3 in the next section. Page 8 of 30

9 3. Equipoise Curve Let us first establish the existence of the asymptotic expansion of E 1 (a) for suitable a. Theorem 3.1. For any finite sequence a (R + ) n with n 2 and Ω n 1 (a) nonzero, there is an infinite sequence α := (α 1, α 2,...) R such that the equipoise curve E 1 (a) has asymptotic expansion ( ) y x 1 + α s x s as x. s=1 Proof. By Theorem 2.3, there is an α 1 R such that E 1 (a) is y = x + α 1 + O(x 1 ) as x. Now assume for some positive integer N that there is a sequence (α 1, α 2,..., α N ) R N such that E 1 (a) is ( ) N y = x 1 + α s x s + f N (x), s=1 with O(f N (x)) = O(x N ) as x. Then P a (x, 1) = S a (x, y) ( n = a r x (x n r 1 + = n a r x (1 n 1 + ) N α s x s + f N (x) s=1 ) r 1 N α s x s s=1 ) r 1 Page 9 of 30

10 + n (r 1)a r x n 2 f N (x) + O(x n N 3 ). Note that O(x n 2 f N (x)) = O(x n N 2 ). Our assumption for N implies that coefficients of powers of x down as far as x n N 1 on the right match the corresponding coefficients in P a (x, 1), so it follows that n (r 1)a r f N (x) = cx N + O(x N 1 ), where the coefficient c is equal to the difference between the coefficients of x n N 2 in P a (x, 1) and in Σ n a r x n 1 (1 + Σ N s=1α s x s ) r 1. The coefficient of f N (x) is nonzero because Ω n 1 (a) is nonzero. Let α N+1 := c/σ n (r 1)a r. Then E 1 (a) is y = x ( 1 + N+1 s=1 The theorem now follows by induction on N. α s x s ) + O(x N 1 ). Under the conditions of Theorem 3.1, the equipoise curve E 1 (a) has an asymptotic expansion with coefficient sequence α = (α 1, α 2,...) as x. Since W 2 (a) = Σ n (r 1)a r, the proof of Theorem 3.1 shows that α N = c N /W 2 (a), where c N is the difference between the coefficients of x N in the expansions P a (x, 1) x n 1 = Σ 1 (a) + Σ 2 (a)x 1 + Σ 3 (a)x = Σ k+1 (a)x k k=0 Page 10 of 30

11 and n a r (1 + N 1 s=1 α s x s ) r 1 := C N,k (a)x k, so c N = Σ N+1 (a) C N,N (a). Of course, we have yet to determine the coefficients C N,k (a), but note immediately that C N,k (a) = 0 for all sufficiently large k. Let d := (d 1,..., d N 1 ) (Z + ) N 1 be a nonnegative integer sequence such that Σ N 1 s=1 sd s = k and Σ N 1 s=1 d s = 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 a term in x k in the expansion of ( 1 + Σs=1 N 1 α s x s) r 1, with coefficient k=0 (r 1)! (r m 1)!d 1!d 2!... d N 1! αd 1 1 α d α d N 1 N 1. For convenience we abbreviate such expressions with the following compact notation for the product α(d) := and the multinomial coefficient ( ) Σ(d) := d N 1 s=1 α ds s m! Π N 1 s=1 d s!, where Σ(d) = m. Thus the coefficient of the term in x k corresponding to d in the expansion of ( 1 + Σ N 1 s=1 α s x s) r 1 becomes ( )( ) r 1 Σ(d) α(d), Σ(d) d Page 11 of 30

12 where the first factor is the binomial coefficient ( ) r 1 m, which by definition is 0 when m > r 1. Let P (k, N 1, m) (Z + ) N 1 be the set of all partitions of k with length ( N 1 and weight m. Then in the expansion of Σ n a r 1 + Σ N 1 s=1 α s x s) r 1 the coefficient of the term in x k corresponding to any particular d P (k, N 1, m) is n ( r 1 m ) a r ( m d ) α(d) = ( ) m α(d)w m+1 (a). d Summing over all partitions in P (k, N 1, m) and all relevant weights m yields C N,k (a), the total coefficient of x k. When k = N we obtain the coefficient C N,N (a) needed for α N. 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 Ω n 1 (a) nonzero, the asymptotic expansion of the equipoise curve E 1 (a) is ( ) y x 1 + α s x s as x, s=1 where the coefficient sequence α := (α 1, α 2,...) R is given by for each N 1, with C N,N (a) = α N = Σ N+1(a) C N,N (a) W 2 (a) N m=2 d P (N,m) ( ) m α(d) W m+1 (a). d Page 12 of 30

13 In particular, when N = 1 we have P (1, m) = so C 1,1 (a) = 0, since its inner sum is empty. This gives α 1 = Σ 2 (a)/ W 2 (a), consistent with Theorem 2.3. Again, when N = 2 the sequence (2) R 1 is the unique partition of 2 with length 1, so P (2, 2) = {(2)} and the inner sum for C 2,2 (a) is ( 2 2) α(2) = α 2 1, whence C 2,2 (a) = α1w 2 3 (a). Then α 2 = (Σ 3 (a) α1w 2 3 (a))/ W 2 (a), again consistent with Theorem 2.3. Substituting here for α 1 and suppressing the argument a to simplify notation yields α 2 = Σ 3W2 2 Σ 2 2W 3. W2 3 When N = 3 we have P (3, 2) = {(1, 1)} and P (3, 3) = {(3, 0)}, so Theorem 3.2 yields α 3 from C 3,3 (a) = 2α 1 α 2 W 3 (a) + α 3 1W 4 (a). Substituting for α 1 and α 2 and suppressing the argument a now yields α 3 = Σ 4W2 4 Σ 3 2W 2 W 4 + 2Σ 3 2W3 2 2Σ 2 Σ 3 W2 2 W 3. W2 5 Evidently continuing this process will yield an expression for any α N 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 α N, but first we introduce some notation. For any integer sequence d (Z + ) N let Σ(d) := N Σ r+1 (a) dr and W(d) := N W r+1 (a) dr. Page 13 of 30

14 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 Q(N) := (d, e) : d, e (Z + ) N, rd r = N, N re r = 2N 2, } N (d r + e r ) = 2N 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 α N in the asymptotic expansion of the equipoise curve E 1 (a) satisfies an identity of the form α N W2 2N 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 α 1 W 2 = Σ 2, 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 α s with 1 s < N. By Theorem 3.2, α N W 2N 1 2 = Σ N+1 W 2N 2 2 N M=2 D P (N,M) ( ) M α(d) W2 2N 2 W M+1. D Page 14 of 30

15 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 α N identity. Each summand is of the form D P (N,M) ( ) M W2 2N M α(d) W2 M 2 W M+1. D Since 2N M = Σ N 1 s=1 (2s 1)D s for each D P (N, M), we have N 1 W2 2N M α(d) = s=1 (α s W 2s 1 2 ) Ds = N 1 s=1 (d,e) Q(s) c(d, e)σ(d)w (e) where the last step is by hypothesis. If (d, e), (d, e ) Q(s) then Σ(d)W(e) Σ(d )W(e ) = Σ(d + d )W(e + e ), so it follows that every term in the expansion of the sum over Q(s), raised to the power D s, 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 sd s and (2s 2)D s respectively, with sum of weights (2s 1)D s. 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 D s Page 15 of 30

16 of Σ N 1 s=1 sd s = N and e is a partition of Σs=1 N 1 (2s 2)D s = 2N 2M, and the sum of weights of d and e is Σs=1 N 1 (2s 1)D s = 2N M. Before we sum over M, recall that each such term is multiplied by W2 M 2 W M+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 α s 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 α 4 W 7 2 is a sum of eight products of Σ k s and W k s, with coefficients as noted: (d, e) Q(4) c(d, e) (d, e) Q(4) c(d, e) ((0, 0, 0, 1), (6, 0, 0, 0)) +1 ((2, 1, 0, 0), (2, 2, 0, 0)) +6 ((1, 0, 1, 0), (4, 1, 0, 0)) 2 ((4, 0, 0, 0), (2, 0, 0, 1)) 1 ((0, 2, 0, 0), (4, 1, 0, 0)) 1 ((4, 0, 0, 0), (1, 1, 1, 0)) +5 ((2, 1, 0, 0), (3, 0, 1, 0)) 3 ((4, 0, 0, 0), (0, 3, 0, 0)) 5 In particular, the term in α 4 W 7 2 with the largest coefficient is 6Σ 2 2Σ 3 W 2 2 W 2 3, and the term independent of W 2 is 5Σ 4 2W 3 3. 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 Page 16 of 30

17 homogeneous polynomial in 2N variables, F N (u 1,..., u N ; v 1,..., v N ) := (d,e) Q(N) c(d, e) N u dr r N s=1 v es s, when evaluated at u r = Σ r+1 (a), v s = W s+1 (a), 1 r, s N, takes the value F N (Σ 2 (a),..., Σ N+1 (a); W 2 (a),..., W N+1 (a)) = α N W 2 (a) 2N 1. From the 2N variables u r, v s (1 r, s N) let us form 2N 1 rational variables: ρ r := u r v r (1 r N) and τ s := v s+1 v 1 (1 s N 1). Then F 1 (u 1 ; v 1 )=u 1 =ρ 1 v 1 and F 2 (u 1, u 2 ; v 1, v 2 )=u 2 v 2 1 u 2 1v 2 =(ρ 2 ρ 2 1)v 2 1v 2, whence F 1 (u 1 ; v 1 ) v 1 = ρ 1 and F 2 (u 1, u 2 ; v 1, v 2 ) v 3 1 = (ρ 2 ρ 2 1)τ 1. A corresponding identity can be obtained for each F N. Indeed if N 2 then N 1 c N,N = α1 N W N+1 + ( ) m α(d) W m+1, d by Theorem 3.2, whence m=2 d P (N,m) F N (u 1,..., u N ; v 1,..., v N ) = u N v 2N 2 1 u N 1 v N 2 1 v N + R N, Page 17 of 30

18 where R N = R N (u 1,..., u N 1 ; v 1,..., v N 1 ) is a polynomial which does not involve the variables u N and v N. Now u N v 2N 2 1 u N 1 v N 2 1 v N = (ρ N ρ N 1 )τ N 1 v 2N 1 1, whence induction on N utilizing Theorem 3.2 and Corollary 3.4 establishes the general identity for F N 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 N 1 ρ dr r and τ(e) := τs es. Also for any partition pair (d, e) Q(N) note that the sequence d (Z + ) N 1 given by d := Ω N 1 (d + e) is a length N 1 partition of N 1, since N 1 rd r = = N (r 1)(d r + e r ) r=2 N rd r + N re r s=1 N (d r + e r ) = 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) : Ω N 1 (d + e) = d }. Then induction establishes Page 18 of 30

19 Corollary 3.5. For any integer N 2, the polynomial F N satisfies the identity F N (u 1,..., u N ; v 1,..., v N ) = c(d, e)ρ(d)τ(ω v1 2N 1 N 1 (d + e)) (d,e) Q(N) = f d (ρ 1,..., ρ N )τ(d ), where In particular we have f d (ρ 1,..., ρ N ) := d P (N 1) (d,e) Q (d ) f (1) = ρ 2 ρ 2 1, c(d, e)ρ(d). f (2,0) = 2ρ 3 1 2ρ 1 ρ 2, f (0,1) = ρ 3 ρ 3 1, f (3,0,0) = 6ρ 2 1ρ 2 ρ 2 2 5ρ 4 1, f (1,1,0) = 5ρ 4 1 3ρ 2 1ρ 2 2ρ 1 ρ 3, f (0,0,1) = ρ 4 ρ 4 1. For each d P (N 1), the polynomial f d in the N variables ρ 1,..., ρ N has coefficients which are a subfamily of the coefficients introduced in Theorem 3.3. Each (d, e) Q(N) determines a unique d = Ω N 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 Ω N 1 (e) = d and e 1 = Σ N 1 d r. 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 f d contains the term c(d, e)ρ N 1, and it follows that f d is of degree N. In particular our earlier calculations show that f d (ρ 1,..., ρ N ) = ρ N ρ N 1 when d = (0,..., 0, 1) P (N 1), and when evaluated at ρ 1 =... = ρ N = 1 this polynomial takes the value 0, so the sum of its coefficients is 0. Then induction establishes Page 19 of 30

20 Corollary 3.6. For any N 2 and each d P (N 1), the polynomial f d (ρ 1,..., ρ N ) is of degree N, and the sum of its coefficients is 0. Since each f d satisfies f d (1,..., 1) = 0, it immediately follows that we have Corollary 3.7. For any integer N 2, the polynomial F N (u 1,..., u N ; v 1,..., v N ) has sum of coefficients equal to 0, and in fact satisfies the identity F N (u 1,..., u N ; u 1,..., u N ) = 0. Page 20 of 30

21 4. Particular Evaluations of the Asymptotic Expansion We shall now consider evaluations of the coefficient sequence α = (α 1, α 2,...) of the asymptotic expansion of the equipoise curve E 1 (a), for particular choices of a (R + ) n. First, note that Σ N+1 (a) = 0 = W N+1 (a) for all N n. This causes no concern if we use Corollary 3.4 to determine α N by evaluating F N at u r = Σ r+1 (a), v s = W s+1 (a), 1 r, s N. On the other hand, it is not immediately obvious how we should use Corollary 3.5 to determine α N when N n, since the rational variable ρ r = u r /v r does not have a stand-alone value when u r = 0, v r = 0. However, for each partition pair (d, e) Q(N) the product ρ(d)τ(ω N 1 (d + e)) contains the factor ρ dr r τ dr+er r 1 = u dr r vr er /v dr+er 1, which takes the value 0 when u r = 0 and v r = 0, since Ω n 1 (a) nonzero ensures that v 1 = W 2 (a) is nonzero. Thus, Corollary 3.5 yields α N as the value of F N /v1 2N 1 when u r = Σ r+1 (a), v s = W s+1 (a), 1 r, s N, by noting that the only products f d τ(d ) that can be nonzero correspond to partitions d P (N 1) with Ω N n (d ) = 0. Example 4.1. If a (R + ) 2 with Ω 1 (a) nonzero, it is trivial to verify that the equipoise curve E 1 (a) is the straight line y = x+α 1, with α 1 = a 1 and α N = 0 for N 2. Example 4.2. If a (R + ) 3 with Ω 2 (a) nonzero, the equipoise curve E 1 (a) is a hyperbola with asymptote y = x + α 1. Corollary 3.5 yields α N as the value of f d τ(d ) for d = (N 1, 0,..., 0) P (N 1), with the evaluation ρ 1 = Σ 2 (a)/ W 2 (a), ρ 2 = Σ 3 (a)/ W 3 (a) and τ 1 = W 3 (a)/ W 2 (a), noting that Page 21 of 30

22 any products of ρ 2 and τ 1 are equal to zero if a 3 = 0. We have α 1 = ρ 1, α 2 = (ρ 2 ρ 2 1)τ 1, α 3 = (2ρ 3 1 2ρ 1 ρ 2 )τ 2 1, α 4 = (6ρ 2 1ρ 2 ρ 2 2 5ρ 4 1)τ 3 1, 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 α N recursively in this case. In fact, with α 1 and α 2 as determined above, we have all later terms given by the recurrence α N = W 3(a) W 2 (a) ( N 1 s=1 α s α N s ) for N 3. The substitution β N := τ 1 α N for N 1 converts the recurrence to a pure convolution β N = N 1 s=1 β s β N s for N 3, corresponding to the classical recurrence satisfied by Catalan numbers, but now with initial conditions β 1 = ρ 1 τ 1 and β 2 = (ρ 2 1 ρ 2 )τ 2 1. 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 F (z) := β 1 z + β 2 z β N z N +... = β r z r, Page 22 of 30

23 and letting Z := τ 1 z, we find that F satisfies F (z) 2 F (z) Z(ρ 1 + ρ 2 Z) = 0, so F (z) = Z(ρ 1 + ρ 2 Z). 2 Binomial series expansion now leads to an explicit closed-form solution for β N, whence N/2 α N = ( τ 1 ) N 1 s=0 ( ( 1) s 2N 2s 1 2N 2s 1 N 2s, s, N s 1 ) ρ N 2s 1 ρ s 2 for N 1. 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 [ ( ) 1 9 α 5 = τ1 4 ρ ( ) 7 ρ 3 9 5, 0, 4 7 3, 1, 3 1ρ ( ) ] 5 ρ 1 ρ 2 2, 5 1, 2, 2 whence α 5 = (14ρ ρ 3 1ρ 2 +6ρ 1 ρ 2 2)τ 4 1. Note by Corollary 3.6 that each f d has coefficient sum zero, so the generalised Catalan numbers have zero alternating sum for N 2 : N/2 s=0 ( ( 1) s 2N 2s 1 2N 2s 1 N 2s, s, N s 1 ) = 0. Alternatively, this identity can be deduced from the quadratic identity for F (z) when ρ 2 1 = ρ 2. Putting Z := ρ 1 τ 1 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. Page 23 of 30

24 Example 4.3. Let us now consider the constant sequence a = 1 (R + ) n in which each term is equal to 1. Then Σ k (1) = ( n k) = Wk (1) for 1 k n, so in this case α 1 = 1 and Corollaries 3.4 and 3.7 imply that α N = 0 for N 2. Hence, as in Example 4.1, the equipoise curve E 1 (1) is the straight line y = x + 1. This is confirmed by noting that P 1 (x, 1) = (x + 1) n x n and S 1 (x, y) = (x n y n )/(x y), so P 1 (x, 1) = S 1 (x, y) holds when y = x + 1. Example 4.4. Let δ := (δ 1, δ 2,..., δ n ) R n be a sequence in which every term satisfies δ r ɛ for some small strictly positive ɛ R +. Then a := 1 + δ (R + ) n is a small perturbation of the constant sequence 1. Let Σ 0 (δ) := 1. Then for each k 1 we have Σ k (1 + δ) = k s=0 ( ) n s Σ s (δ) and W k (1 + δ) = k s ( ) n + W k (δ). k It is convenient to scale the functions Σ k and W k by dividing by Σ k (1) = W k (1) = ( n k) when 1 k n : for all a (R + ) n we define Σ k(a) := Σ k (a)/ Σ k (1) and W k (a) := W k (a)/w k (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 Σ k(1 + δ) = k s=0 ( ) k Σ s s(δ) and Wk (1 + δ) = 1 + Wk (δ). Page 24 of 30

25 We keep n fixed and let ɛ 0 +, so O(Σ k (δ)) = O(ɛk ) and O(Wk (δ)) = O(ɛ). In particular, α 1 = 1 + 2Σ 1(δ) W2 (δ) + O(ɛ 2 ). For any integer s 0, put ( )/ ( ) ( )/ ( ) n n n 2 s + 2 λ s := = s s 2 and for any d (Z + ) N 1 define λ(d) := Πs=1 N 1 λ ds s. Now evaluating the coefficient α N at 1 + δ using Corollaries 3.4 and 3.5 is convenient so long as we know f d explicitly for each d P (N 1). We have O(f d ) = O(ɛ) because ρ r = 1 + O(ɛ) for 1 r n 1 and f d has zero coefficient sum by Corollary 3.6. As τ s = λ s + O(ɛ) for 1 s n 2, it follows for N 2 that α N = f d (ρ 1,..., ρ N )λ(d ) + O(ɛ 2 ) where f d (ρ 1,..., ρ N ) = (d,e) Q (d ) c(d, e) d P (N 1) ( NΣ 1(δ) + N [ d r Σ 1 (δ) Wr+1(δ) ]) + O(ɛ 2 ). On the other hand, using Theorem 3.2 to evaluate α N at 1 + δ yields α 1 as Page 25 of 30

26 above and for N 2 yields α N via the recurrence N 2 α N = (s + 1)λ s α N s Nλ N 1 (Σ 1(δ) W2 (δ)) s=1 + λ N 1 ( Σ 1 (δ) W N+1(δ) ) + O(ɛ 2 ). It follows by induction for N 2 that α N 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 A n 1 (δ) = 0, so δ r = 0 for 1 r n 1 and δ n ɛ. Then Σ 1(δ) = δ n /n and Wk (δ) = kδ n/n for 2 k n, so α 1 = 1 and α N = 0 for N 2. This is confirmed directly by checking that P 1+δ (x, 1) = (x + 1) n x n + δ n (x + 1) n 1 and S 1+δ (x, y) = (x n y n )/(x y) + δ n y n 1 are equal precisely when y = x + 1. Page 26 of 30

27 5. Selected Inequalities To conclude, let us briefly sample some of the inequalities between the polynomials P a (x, 1) and S a (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, P 1 (1, 1) = 2 n 1 and S 1 (1, 3) = (3 n 1)/2. The point (1, 3) lies above the equipoise curve, so is in the S-region, and Theorem 2.2 implies 2 n 1 < 3n 1. 2 Indeed, the lines y = x + 2 and y = x + 1 lie, respectively, in the S-region and 2 the P -region, so for x > 0 we have ( x + 1 ) n x n < (x + 1)n x n 2 2 < (x + 2)n x n. 4 These inequalities would usually be deduced from the convexity of y = x n, and actually hold for all x R when n is even. Since α 1 = 1 and α N+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 ( x x N ) n x n 1 1 x N < (x + 1) n x n < ( x x N ) n x n x N Page 27 of 30

28 Once again these inequalities could be deduced from convexity of y = x n, 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 ɛ R + such that δ r ɛ for 1 r n. Let us consider the special case in which δ 1 = ɛ, δ n = ɛ and δ r = 0 for 2 r n 1. Then P 1+δ (x, 1) = (x + 1) n x n ɛ 2 (x + 1) n 2 S 1+δ (x, y) = yn x n y x + ɛ(yn 1 x n 1 ). In this case Σ 1(δ) = 0, Σ 2(δ) = 2ɛ/n(n 1), Σ k (δ) = 0 for 3 k n, and (δ) = kɛ/n for 2 k n. From Example 4.4 we have W k α 1 = 1 2 n ɛ+o(ɛ2 ), α 2 = n 2 (n + 1)(n 2) 3n ɛ+o(ɛ2 ), α 3 = ɛ+o(ɛ 2 ). 18n Then α 2 > 0 and α 3 < 0, so the curves y = x + α 1 and y = x + α 1 + α 2 x 1 lie, respectively, below and above the equipoise curve for sufficiently large x. Hence (x + α 1 ) n x n + ɛ [ (x + α 1 ) n 1 x n 1] α 1 and < (x + 1) n x n ɛ 2 (x + 1) n 2 ( ) x + α1 + α 2 n < x x n [ ( α 1 + α + ɛ x + α α ) n 1 2 x n 1], x x Page 28 of 30

29 where α 1 and α 2 take the exact values ( n ) 2 ɛ 2 α 1 = ( n and α 2 = 2) + (n 1)ɛ ( n ) 3 (n 2)ɛ 2 α1 2 [( n ( 3) + n 1 ) ] 2 ɛ ( n. 2) + (n 1)ɛ Case 5.3. a := (a, b, c) (R + ) 3 with Ω 2 (a) = (b, c) (0, 0). As shown in Example 4.2, for N 1 each α N is a function of ρ 1, ρ 2 and τ 1 in this case. If c = 0, we easily verify that α 1 = a and α N = 0 for N 2. Now suppose c > 0. Then ρ 1, ρ 2 and τ 1 have stand-alone values, and α 2 = 0 precisely when ρ 2 = ρ 2 1. But then α N = 0 for N 2, by the previously noted alternating sum identity for the generalised Catalan numbers. If ρ 2 > ρ 2 1 then α 2 > 0 and the summation identity for α N in Example 4.2 implies ( 1) N α N > 0, so α N alternates in sign for N 2. Similarly if ρ 2 < ρ 2 1 then α 2 < 0 and α N alternates in sign for N 2. As in Case 5.2 above we can deduce relevant inequalities. In particular, if ρ 2 > ρ 2 1 then for sufficiently large x we have ax 2 + bx(x + α 1 ) + c(x + α 1 ) 2 < (x + a)(x + b)(x + c) x 3 ( < ax 2 + bx x + α 1 + α ) ( 2 + c x + α 1 + α ) 2 2, x x where α 1 and α 2 have their exact values, given explicitly in Example 4.2. Page 29 of 30

30 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: /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), Page 30 of 30