Enhancement of the Cauchy-Schwarz Inequality and Its Implications for Numerical Radius Inequalities

TL;DR

This paper improves the classical Cauchy-Schwarz inequality using a function-based approach and applies the result to derive tighter bounds for the numerical radius of operators.

Contribution

It introduces a novel inequality enhancement for Cauchy-Schwarz and utilizes it to obtain improved numerical radius inequalities.

Findings

Derived a new inequality generalizing Cauchy-Schwarz.

Established tighter upper bounds for the numerical radius.

Applied the inequality to operator theory contexts.

Abstract

In this article, we establish an improvement of the Cauchy-Schwarz inequality. Let $x, y \in \mathcal{H},$ and let $f: (0,1) \rightarrow \mathbb{R}^+$ be a well-defined function, where $\mathbb{R}^+$ denote the set of all positive real numbers. Then \[|\langle x, y \rangle|^2 \leq \frac{f(t)}{1+f(t)} \|x\|^2 \|y\|^2 + \frac{1}{1+ f(t)} |\langle x, y \rangle | \|x\|\|y\|. \] We have applied this result to derive new and improved upper bounds for the numerical radius.