In this paper, we find new upper bounds for the Berezin number of the product of bounded linear operators defined on reproducing kernel Hilbert spaces. We also obtain some interesting upper bounds concerning one operator, the upper bounds obtained here refine the existing ones. Further, we develop new lower bounds for the Berezin number concerning one operator by using their Cartesian decomposition. In particular, we prove that ber(A) ≥ 1/ √ 2 ber( ℜ(A) ± ℑ(A)) , where ber(A) is the Berezin number of the bounded linear operator A.