Refined upper bounds for the numerical radius via weighted operator means

Abstract

We establish new upper bounds for the numerical radius of bounded linear operators on a complex Hilbert space by introducing weighted geometric means of the modulus of an operator and its adjoint. This approach yields a family of inequalities that extend and strictly refine several well-known bounds due to Kittaneh and Bhunia--Paul, except in normal or degenerate cases. Further improvements are obtained by interpolating numerical radius estimates with spectral radius bounds, leading to a hierarchy of hybrid inequalities that provide sharper control for non-normal operators. Applications to $2\times2$ operator matrices are presented, and the equality cases are completely characterized, revealing strong rigidity phenomena. Explicit examples are included to illustrate the strictness of the new bounds.