Some generalized numerical radius inequalities involving Kwong functions

TL;DR

This paper establishes new numerical radius inequalities for positive semidefinite matrices using Hadamard products and Kwong functions, extending existing bounds and providing generalized inequalities.

Contribution

It introduces novel numerical radius inequalities involving Kwong functions and positive semidefinite matrices, broadening the scope of matrix analysis techniques.

Findings

Derived bounds for the numerical radius involving Hadamard products.

Established inequalities connecting functions of matrices with their spectra.

Extended classical inequalities to more general classes of functions and matrices.

Abstract

We prove several numerical radius inequalities involving positive semidefinite matrices via the Hadamard product and Kwong functions. Among other inequalities, it is shown that if $X$ is an arbitrary $n\times n$ matrix and $A,B$ are positive semidefinite, then \begin{align*} \omega(H_{f,g}(A))\leq k\, \omega(AX+XA), \end{align*} which is equivalent to \begin{align*} \omega\big(H_{f,g}(A,B)\pm H_{f,g}(B,A)\big)\leq k'\,\left\{\omega((A+B)X+X(A+B))+\omega((A-B)X-X(A-B))\right\}, \end{align*} where $f$ and $g$ are two continuous functions on $(0,\infty)$ such that $h(t)={f(t)\over g(t)}$ is Kwong, $k=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)}\right\}$ and $k'=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)\cup\sigma(B)}\right\}$.