Berezin number inequalities for Hilbert space operators

Abstract: In this paper, by using of the definition Berezin symbol, we show some Berezin number inequalities. Among other inequalities, it is shown that if $A, B, X\in{\mathbb{B}}(\mathscr H)$, then $$\mathbf{ber}(AX\pm XA)\leqslant \mathbf{ber}^{\frac{1}{2}}\left(A^A+AA^\right)\mathbf{ber}^{\frac{1}{2}}\left(X^X+XX^\right)$$ and $$\mathbf{ber}^2(A^*XB)\leqslant|X|^2\mathbf{ber}(A^*A)\mathbf{ber}(B^*B).$$