For a bounded linear operator $A$ on a functional Hilbert space $mathcal{H}left( Omegaright) $, with normalized reproducing kernel $widehat {k}_{eta}:=frac{k_{eta}}{leftVert k_{eta}rightVert _{mathcal{H}}},$ the Berezin symbol and Berezin number are defined respectively by $widetilde{A}left( etaright) :=leftlangle Awidehat{k}_{eta},widehat{k}_{eta}rightrangle _{mathcal{H}}$ and $mathrm{ber}(A):=sup_{etainOmega}leftvert widetilde{A}{(eta)}rightvert .$ A simple comparison of these properties produces the inequality $mathrm{ber}% left( Aright) leqfrac{1}{2}left( leftVert ArightVert_{mathrm{ber}}+leftVert A^{2}rightVert _{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