Tracing the Origins and Generalizations of a Class of Moving Point Inequalities

Abstract

This research begins with the Guggenheimer inequality and systematically reviews its developmental progress, which has generally undergone stages of degree generalization, form strengthening, and weighted generalization. Following this, inequalities structurally similar to the Guggenheimer inequality have continuously emerged. The Hungarian Mathematical Competition of 2015–2016 presented an inequality with a similar structure. A higher-degree generalization of this inequality leads to a more general form of moving point inequality. By combining the structure of the strengthened generalization of the Guggenheimer inequality, five conjectures with similar structures are proposed.