Some Upper Bounds for the Berezin Number of Hilbert Space Operators

In this paper, we obtain some Berezin number inequalities based on the definition of Berezin symbol. Among other inequalities, we show that if A, B be positive definite operators in B(H), and A#B is the geometric mean of them, then ber(2)(A#B) <= ber(A(2)+B-2/2)-1/2 inf (lambda is an element of Omega )zeta((k) over cap (lambda)), where zeta((k) over cap (lambda)) = <(A - B)(k) over cap (lambda),(k) over cap (lambda)>(2), and (k) over cap (lambda) is the normalized reproducing kernel of the space H for lambda belong to some set Omega.