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  Volume 3, Issue 5, Article 84
 
Asymptotic Expansion of the Equipoise Curve of a Polynomial Inequality
    Authors: Roger B. Eggleton, William P. Galvin,  
    Keywords: Polynomial inequality, Catalan numbers.  
    Date Received: 28/08/02  
    Date Accepted: 08/11/02  
    Subject Codes:

26D05,26C99,05A99

  
    Editors: Hillel 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 $ n=3,$ 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.

         
       
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