This present work aims to ameliorate the celebrated Cauchy-Schwarz inequality and provide several new consequences associated with numerical radius upper bounds of Hilbert space operators. More precisely, for arbitrary a, b ? H 0, we show that |?a,b?|2 1 + ?a??b?|?a, b?| ?/?+1 ?a?2?b?2 ?a?2?b?2. As a consequence, refine generalize some Kittaneh?s results in [A an estimate Frobenius companion ma...