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

Abstract

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 ? H and ? ? 0, we show that |?a,b?|2 ? 1 ? + 1 ?a??b?|?a, b?| + ?/?+1 ?a?2?b?2 ? ?a?2?b?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 ? B(H) and ? ? 0, we show the following sharp upper bound w2 (B*A) ? 1/2?+2 ?|A|2 + B|2?w(B*A)+ ?/2?+2 ?|A|4 + |B?4, with equality holds when A=B= (0100). It is also worth mentioning here that some specific values of ? ? 0 provide more accurate estimates for the numerical radius. Finally, some related upper bounds are also provided.