Scientific Paper: Refinement of the Cauchy-Schwartz inequality

Motivated by the results previously reported by our partner researcher, his present work aims to develop new upper bounds for the numerical radius of Hilbert space operators by introducing further improvements to the well-known Cauchy–Schwarz inequality. In particular, a novel Lemma (3.1) is proposed and utilized to extend and generalize several existing vector and numerical radius inequalities, as well as previously established extensions of the Cauchy–Schwarz inequality. Specifically, inequalities (2.5), (2.8), and (1.6) are generalized by (4.3), (4.1), and (4.2), respectively.

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