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 λ∈Ω ζ(k λ), where ζ(k λ) = (A − B)k λ ,k λ 2 , andk λ is the normalized reproducing kernel of the space H for λ belong to some set Ω. A(λ) = Ak λ (z),k λ (z) ,
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