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 ber2(A]B) ber A2 + B2 2 ! 1 2 inf 2 (ˆk); where (ˆk) = h(A B)ˆk;ˆki2; and ˆk is the normalized reproducing kernel of the space H for belong to some set :
