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...

The aim of this paper is to provide new upper bounds ω(T), which denotes the numerical radius a bounded operator T on Hilbert space (H,⟨·,·⟩). We show Aczél inequality in terms |T|. Next, we give certain inequalities about A-numerical ωA(T) and A-operator seminorm ∥T∥A an T. also present several results related 2×2 block matrices semi-Hilbert operators, by using symmetric matrices.