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

Abstract

In this article, we establish an improvement of the Cauchy-Schwarz inequality. Let $x,y\in \mathcal{H},$ and let $f:(0,1)\to {\mathbb{R}}^{+}$ 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.