On Berezin radius inequalities via Cauchy-Schwarz type inequalities

Abstract

The Berezin transform $\widetilde{A}$ and the Berezin radius of an operator $A$ on the reproducing kernel Hilbert space over some set $\Omega$ with the reproducing kernel $k_{\lambda}$ are defined, respectively, by \[ \widetilde{A}(\lambda)=\left\langle {A}\widehat{{k}}_{\lambda},\widehat{{k}% }_{\lambda}\right\rangle ,\ \lambda\in\Omega\text{ and }\mathrm{ber}% (A):=\sup_{\lambda\in\Omega}\left\vert \widetilde{A}{(\lambda)}\right\vert . \] We study some new inequalities by using this bounded function $\widetilde{A}, $ involving refinements of some Berezin radius inequalities for operators acting on the reproducing kernel Hilbert space.