Finding best possible constant for a polynomial inequality

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 $a^3+b^3+c^3+k(a^{2}b+b^{2}c+c^{2}a)-(k+1)(a{b}^2+b{c}^2+c{a}^2)\ge 0$ 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.

Proceedings of the 20th Asian Technology Conference in Mathematics (Leshan, China, 2015) 178-187