Geometric means inequalities and their extensions to Lie groups

Abstract

This dissertation has two main parts.

1. After reviewing the Riemannian structure of the space of n n positive de nite matrices, Pn, and the geometric mean in terms of geodesic, t-geometric mean, we present some inequalities of Dinh, Ahsani, and Tam [15] involving t-geometric mean in the context of Pn. Some very recent geometric inequalities of Lemos and Soares [24] are also presented.

2. After reviewing some preliminary materials of Lie groups and Lie algebras, we obtain extensions of the inequalities of Lemos and Soares in the context of semisimple Lie groups.