The Berezin symbol A of an operator A on the reproducing kernel Hilbert space H (Ω) over some set Ω with the reproducing kernel k ξ is defined by ˜ A(ξ) = A k ξ k ξ , k ξ k ξ , ξ ∈ Ω. The Berezin number of an operator A is defined by ber(A) := sup ξ∈Ω A(ξ). We study some problems of operator theory by using this bounded function A, including treatments of inner product inequalities via convex functions for the Berezin numbers of some operators. We also establish some inequalities involving of the Berezin inequalities.