In this paper we find new estimate for the numerical radius of a given matrix, and we prove that, this estimate is better than any estimate for the numerical radius. We present also new bounds for the zero of polynomials by using new estimate for the numerical radius of a companion matrix of a given polynomial and matrix inequalities.

Multiplicative preservers of C-numerical ranges and radii on certain groups and semigroups of complex n × n matrices are characterized. The general and special linear groups are considered, as well as the semigroups of matrices having ranks not exceeding k, with k fixed in advance. For a fixed C, it turns out that typically the multiplicative preservers of the C-numerical range (or radius) have...

In this article, we prove an inner product inequality for Hilbert space operators. This inequality, then, is utilized to present a general numerical radius using convex functions. Applications of the new results include obtaining forms that generalize and extend some well known in literature, with application newly defined generalized radius. We emphasize approach followed article different fro...