Matrix Inequalities for Sector Matrices: A Comprehensive Review of Developments and Generalizations

Abstract

In recent years, the notion of sector matrices, which are natural extensions of positive definite Hermitian matrices, has played a pivotal role in the development of various matrix inequalities. In this review article, we present a unified perspective on key advances in this area, from the early results of Lin and Drury on matrix inequalities to the recent work of Khosravi, Malekinejad, and their collaborators concerning operator means and the role of the sectorial angle in generalized constants. In particular, we focus on how classical inequalities, such as P\'{o}lya type inequalities and other operator mean related inequalities, are extended and proved within the framework of sector matrices. We also examine in detail the monotonicity and order reversing properties of operator means under the real part mapping, as well as the sharp appearance of the factor \(\sec^2\alpha\) in such generalizations. By consolidating existing results, this article aims to provide a coherent reference and outline promising directions for future research in this active field.