Refined and generalized numerical radius inequalities via a sharpened Buzano inequality

Abstract

This paper refines and generalizes several classical numerical radius inequalities for bounded linear operators on Hilbert spaces. We first obtain a generalization of the Buzano inequality, which in the special case α > 2, becomes a strict refinement of the classical bound. This refined inequality serves as a key tool throughout the paper. Using this together with the Hölder–McCarthy inequality and convexity arguments, we derive new bounds for expressions of the form Aα X Bβ and for powers of a single operator, thereby refining results of Sattari, Moslehian, and Yamazaki, as well as a recent work by Bhunia. Among the main results, we prove a chain of inequalities connecting the Cartesian decomposition to the numerical radius: 1 4 ‖A∗ A + A A∗‖ ≤ 3 √4 8 (‖A + A∗‖6 + ‖A − A∗‖6)1/3 ≤ w2(A). In the self-adjoint case, the middle term is shown to be strictly larger than a comparable bound due to Bhunia and Paul, providing a sharper estimate. In the general case, an example has been given that shows that our bound is better, therefore we have conjectured for it to hold in general.