Improving the Cauchy-Schwarz inequality

TL;DR

This paper explores an improved version of the Cauchy-Schwarz inequality, demonstrating its advantages in singular cases and applications, and highlighting the role of projections in enhancing inequality relations.

Contribution

It introduces a variant of the Cauchy-Schwarz inequality that performs better in singular situations and discusses the impact of projection operators on these inequalities.

Findings

The variant works effectively in singular cases where the original inequality fails.

Applications show improved bounds in uncertainty relations.

Projection operators can modify and enhance the inequality relations.

Abstract

We highlight overlap as one of the simplest inequalities in linear space that yields a number of useful results. One obtains the Cauchy-Schwarz inequality as a special case. More importantly, a variant of it is seen to work desirably in certain singular situations where the celebrated inequality appears to be useless. The basic tenet generates a few other interesting relations, including the improvements over certain common uncertainty bounds. Role of projection operators in modifying the Cauchy-Schwarz relation is noted. Selected applications reveal the efficacy.