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A Polynomial Inequality Generalising an Integer Inequality  
 
  Authors: Roger B. Eggleton, William P. Galvin,  
  Keywords: Polynomial inequality, Sums of Products of Digits, Bernoulli inequality.  
  Date Received: 01/05/02  
  Date Accepted: 07/06/02  
  Subject Codes:

26D15,26C99

  
  Editors: Hillel Gauchman,  
 
  Abstract:

For any $ mathbf{a}:=(a_{1},a_{2},dots ,a_{n})in (mathbb{R}^{+})^{n},$ we establish inequalities between the two homogeneous polynomials $ Delta P_{%% mathbf{a}}(x,t):=(x+a_{1}t)(x+a_{2}t)cdots (x+a_{n}t)-x^{n}$ and $ S_{%% mathbf{a}}(x,y):=a_{1}x^{n-1}+a_{2}x^{n-2}y+cdots +a_{n}y^{n-1}$ in the positive orthant $ x,y,tin mathbb{R}^{+}.$ Conditions for $ Delta P_{%% mathbf{a}}(x,t)leq tS_{mathbf{a}}(x,y)$ yield a new proof and broad generalization of the number theoretic inequality that for base $ bgeq 2$ the sum of all nonempty products of digits of any $ min mathbb{Z}^{+}$ never exceeds $ m,$ and equality holds exactly when all auxiliary digits are $ %% b-1.$ Links with an inequality of Bernoulli are also noted. When $ ngeq 2$ and $ mathbf{a}$ is strictly positive, the surface $ Delta P_{mathbf{a}%% }(x,t)=tS_{mathbf{a}}(x,y)$ lies between the planes $ y=x+tmax {a_{i}:1leq ileq n-1}$ and $ y=x+tmin {a_{i}:1leq ileq n-1}.$ For fixed $ tin mathbb{R}^{+},$ we explicitly determine functions $ alpha ,beta ,gamma ,delta $ of $ mathbf{a}$ such that this surface is $ %% y=x+alpha t+beta t^{2}x^{-1}+O(x^{-2})$ as $ xrightarrow infty ,$ and $ %% y=gamma t+delta x+O(x^{2})$ as $ xrightarrow 0+.$;


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