Finding best possible constant for a polynomial inequality

TL;DR

This paper introduces a general algorithm, implementable via computer algebra systems, for determining the optimal constant in polynomial inequalities involving multiple variables, exemplified by a specific inequality with a parameter.

Contribution

A novel, general algorithm for computing the best possible constant in multi-variable polynomial inequalities is proposed, facilitating automation with computer algebra tools.

Findings

Algorithm successfully finds the optimal constant for the example inequality.

Method can be implemented in Maple and similar computer algebra systems.

Applicable to a wide range of polynomial inequality problems.

Abstract

Given a multi-variant polynomial inequality with a parameter, how to find the best possible value of this parameter that satisfies the inequality? For instance, find the greatest number $k$ that satisfies $ a^3+b^3+c^3+ k(a^2b+b^2c+c^2a)-(k+1)(ab^2+bc^2+ca^2)\geq 0 $ for all nonnegative real numbers $ a,b,c $. Analogues problems often appeared in studies of inequalities and were dealt with by various methods. In this paper, a general algorithm is proposed for finding the required best possible constant. The algorithm can be easily implemented by computer algebra tools such as Maple.