  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.
