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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(A]B) ≤ ber ( A2 + B2 2 ) − 1 2 inf λ∈Ω ζ(k̂λ), where ζ(k̂λ) = 〈(A − B)k̂λ, k̂λ〉, and k̂λ is the normalized reproducing kernel of the space H for λ belong to some set Ω.

Some upper bounds for the Berezin number of Hilbert space operators