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In this paper, several refinements of the Berezin number inequalities are obtained. We generalize inequalities involving powers of the Berezin number for product of two operators acting on a reproducing kernel Hilbert space H = H (Ω) and also improve them. Among other inequalities, it is shown that if A,B ∈ B(H ) such that |A|B = B * |A| , f and g are nonnegative continuous functions on [0,∞) satisfying f (t)g(t) = t (t 0) , then ber p (AB) r p (B)doi:10.7153/jmi-2019-13-79 fatcat:b5ti6mbe7jekxo22dz5c7oagwq