New berezin symbol inequalities for operators on the reproducing kernel hilbert space

We use Kittaneh and Manasrah inequality and Kian’s functional calculus method to prove some new inequalities for Berezin symbols and Berezin numbers of operators. In particular, we prove that ( ber f (A)2) (f(A) p ≥ ber +f )(A)q p q for all self-adjoint operators A on the reproducing kernel Hilbert space H (Ω) with spectrum in J ⊂ (−∞,+∞) and all continuous nonnegative functions f defined on J. We also prove new upper and lower bounds for Berezin numbers of reproducing kernel Hilbert space operators. Among our results, we prove that if A: H (Ω) → H (Ω) is a bounded pseudo-hypornormal operator on the reproducing kernel Hilbert space H (Ω), then for all non-negative non-decreasing pseudo-operator convex function f on [0,∞), we have ⎛ ⎞ ⎛ ⎞ f (ber(A)) ≥12 ∥f ⎝|A| ⎠ + f ⎝|A∗ | ⎠ 1 +ξ|A|2 8 1 +ξ|A|2, ∥ 8 Ber where ‖.‖Ber denotes the Berezin norm of operator