Some generalized numerical radius inequalities involving Kwong functions

Year 2019, Volume: 48 Issue: 4, 951 - 958, 08.08.2019

Mojtaba Bakherad

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
\[ \omega(H_{f,g}(A))\leq k\, \omega(AX+XA), \]
 which is equivalent to
\[\omega\big(H_{f,g}(A,B)\pm H_{f,g}(B,A)\big)\leq k'\,\left\{\omega((A+B)X+X(A+B))+\omega((A-B)X-X(A-B))\right\},\]
 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 $k'=\max\left\{{f(\lambda)g(\lambda)\over \lambda}: {\lambda\in\sigma(A)\cup\sigma(B)}\right\}$.

Keywords

numerical radius, Hadamard product, operator monotone, Kwong function

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