Refinement of the Cauchy-Schwartz inequality with refinements and generalizations of the numerical radius type inequalities for operators

Motivated by the results previously reported, the current work aims at developing new numerical radius upper bounds of Hilbert space opera- tors by offering new improvements to the well-known Cauchy-Schwarz inequal- ity. In particular, a novel Lemma (3.1) is given, which is utilized to further generalize several vector and numerical radius type inequalities, as well as pre- viously given extensions of the Cauchy-Schwartz inequality. Specifically, (2.5) (2.8) (1.6) have been generalized by (4.3) (4.1) (4.2)