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

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 $$\mathrm{\mid }⟨x,y⟩{\mathrm{\mid }}^2\le \frac{f(t)}{1+f(t)}\mathrm{\parallel }x{\mathrm{\parallel }}^2\mathrm{\parallel }y{\mathrm{\parallel }}^2+\frac{1}{1+f(t)}\mathrm{\mid }⟨x,y⟩\mathrm{\mid }\mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }\mathrm{.}$$ We have applied this result to derive new and improved upper bounds for the numerical radius.