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 $k'=\max\left\{ { f(\lambda)g(\lambda)\over \lambda}: { \lambda\in\sigma(A)\cup\sigma(B)}\right\}$.
Categories: math.FA