A refinement of the Cauchy-Schwarz inequality accompanied by new numerical radius upper bounds

This present work aims to ameliorate the celebrated Cauchy-Schwarz inequality and provide several new consequences associated with the numerical radius upper bounds of Hilbert space operators. More precisely, for arbitrary a , b is an element of H and alpha >= 0, we show that |2 <= 1 alpha + 1 parallel to a parallel to parallel to b parallel to |&| + alpha + 1 parallel to a parallel to 2 parallel to b parallel to 2 alpha <= parallel to a parallel to 2 parallel to b parallel to 2. As a consequence, we provide several new upper bounds for the numerical radius that refine and generalize some of Kittaneh's results in [A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math. 2003;158:11-17] and [Cauchy-Schwarz type inequalities and applications to numerical radius inequalities. Math. Inequal. Appl. 2020;23:1117-1125], respectively. In particular, for arbitrary A , B is an element of B(H) and alpha >= 0, we show the following sharp upper bound w(2) (B*A) <= 1/2 alpha+2 parallel to vertical bar A vertical bar(2) + vertical bar B-2 vertical bar omega + alpha , 2 alpha + 2/2 alpha parallel to vertical bar A vertical bar(4)+ vertical bar B vertical bar parallel to(4) with equality holds when A = B provide more accurate estimates for the numerical radius. Finally, some related upper bounds are also provided. It is also worth mentioning here that some specific values of alpha >= 0 provide more accurate estimates for the numerical radius. Finally, some related upper bounds are also provided.