Suppose B(H (Ω)) is the set of all bounded linear operators acting on a reproducing kernel Hilbert space H (Ω). Applying the positivity criteria of 2×2 block matrices, we develop several new upper bounds for the Berezin number of operators in B(H (Ω)) involving Berezin norm, which are better than the earlier ones. Among other results, we obtain that if T,S ∈ B(H (Ω)) and 0 <α< 1, then