Asymptotic Expansion of the Equipoise Curve of a Polynomial Inequality

Volume 3, Issue 5, 2002

Article 84

ASYMPTOTIC EXPANSION OF THE EQUIPOISE CURVE OF A POLYNOMIAL INEQUALITY

DEPARTMENT OF MATHEMATICS
ILLINOIS STATE UNIVERSITY
NORMAL, IL 61790-4520, USA.
E-Mail: roger@math.ilstu.edu

SCHOOL OF MATHEMATICAL AND PHYSICAL SCIENCES
UNIVERSITY OF NEWCASTLE
CALLAGHAN, NSW 2308
AUSTRALIA
E-Mail: mmwpg@cc.newcastle.edu.au

Received 28 August, 2002; Accepted 8 November, 2002.
Communicated by: H. Gauchman

ABSTRACT.

For any $\mathbf{a}:= (a_1,a_2, ... ,a_n) \in (\mathbb{R}^+)^{\mathit{n}},$ define $\Delta P_{\mathbf{a}}(x,t):=(x+a_1t)(x+a_2t) ... (x+a_nt)-x^n$ and $%% S_{\mathbf{a}}(x,y):= a_1x^{n-1}+ a_2x^{n-2}y + ... + a_ny^{n-1}.$ The two homogeneous polynomials $\Delta P_{\mathbf{a}}(x,t)$ and $tS_{\mathbf{a}%% }(x,y)$ are comparable in the positive octant $x,y,t \in \mathbb{R}^+.$ Recently the authors [2] studied the inequality $\Delta P_{\mathbf{a}%% }(x,t) > tS_{\mathbf{a}}(x,y)$ and its reverse and noted that the boundary between the corresponding regions in the positive octant is fully determined by the equipoise curve $\Delta P_{\mathbf{a}}(x,1)=S_{\mathbf{a}}(x,y).$ 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 where the coefficients involve a type of generalised Catalan number, and the case where $\mathbf{a}=\mathbf{1+\delta}$ is a sequence in which each term is close to 1. A selection of inequalities implied by these results completes the paper.

Key words:
Polynomial inequality, Catalan numbers.

2000 Mathematics Subject Classification:
26D05, 26C99, 05A99.

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