Advanced refinements of Berezin number inequalities

For a bounded linear operator $A$ on a functional Hilbert space $\mathcal{H}\left( \Omega\right) $, with normalized reproducing kernel $\widehat {k}_{\eta}:=\frac{k_{\eta}}{\left\Vert k_{\eta}\right\Vert _{\mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by $\widetilde{A}\left( \eta\right) :=\left\langle A\widehat{k}_{\eta},\widehat{k}_{\eta}\right\rangle _{\mathcal{H}}$ and $\mathrm{ber}(A):=\sup_{\eta\in\Omega}\left\vert \widetilde{A}{(\eta)}\right\vert .$ A simple comparison of these properties produces the inequality $\mathrm{ber}% \left( A\right) \leq\frac{1}{2}\left( \left\Vert A\right\Vert_{\mathrm{ber}}+\left\Vert A^{2}\right\Vert _{\mathrm{ber}}^{1/2}\right) $ (see [17]). In this paper, we prove further inequalities relating them, and also establish some inequalities for the Berezin number of operators on functional Hilbert spaces