A note on Lin’s and numerical radii inequalities and application in portfolio optimization

Abstract

We consider Lin's generalizations of Reid's and Halmos' inequalities via polar decomposition approach and establish several numerical radii inequalities for the product of two bounded linear operators on a Hilbert space. The established inequalities are a significant extension of Lin's inequalities. We then explore the applications of the inequalities in finance where when constructing an investment portfolio, investors often need to allocate capital across different assets to achieve a desired risk-return profile. Lin's inequalities can be used to ensure that the portfolio weights satisfy certain constraints, such as minimum and maximum exposure to particular asset classes or sectors.