Berezin number inequalities for operators

The Berezin transform à of an operator A , acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by à ( λ ) = 〉 Aǩ λ , ǩ λ 〈 ( λ ∈ Ω), where 3pt _λ = k_λ \| k_λ \| is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ${ber}{(}A) = { }λ ∈ Ω | { A(λ )} | = { }λ ∈ Ω | { {A{{{{3pt} k} }_λ },{{{{3pt} k} }_λ }} } |$ . In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A , B , X are bounded linear operators on a Hilbert space ℋ, then $${ber}(AX ± XA) {be}{{r}^{{1 2}}}( {A*A + AA*} ){be}{{r}^{{1 2}}}( {X*X + XX*} )$$ and $${be}{{r}^2}({A^*}XB) {\| X \|^2}{ber}({A^*}A){ber}({B^*}B).$$ We also prove the multiplicative inequality $${ber}(AB){ber}(A){ber}(B)$$