Some extensions of Berezin number inequalities on operators

We establish some upper bounds for Berezin number inequalities including inequalities for 2×2 operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if T=[0XY0], then berr(T)≤2r−2(ber(f2r(|X|)+g2r(|Y∗|))+ber(f2r(|Y|)+g2r(|X∗|)))−2r−2inf‖(kλ1,kλ2)‖=1η(kλ1,kλ2), where X, Y are bounded linear operators on a Hilbert space ℋ=ℋ(Ω), r≥1, f, g are nonnegative continuous functions on [0,∞) satisfying the relation f(t)g(t)=t (t∈[0,∞)) and η(kλ1,kλ2)=(⟨(f2r(|X|)+g2r(|Y∗|))kλ2,kλ2⟩12−⟨(f2r(|Y|)+g2r(|X∗|))kλ1,kλ1⟩12)2.