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A functional Hilbert space is defined as the Hilbert space K of complex-valued functions defined on a set Θ. In this space, the evaluation functionals ψ ε ( h ) = h ( ε ) , for ε ∈ Θ , are continuous on K . In this paper, we introduce the mean-Berezin norm ∥ ⋅ ∥ ρ t , defined as follows: ∥ G ∥ M ρ t = sup κ , ξ ∈ Θ { ( | 〈 G k ˆ κ , k ˆ ξ 〉 | ) ρ t ( | 〈 G ∗ k ˆ κ , k ˆ ξ 〉 | ) } , for  0 ⩽ t ⩽ 1 , where G ∈ O ( K ) and ρ t represents an interpolation path of a symmetric mean ρ . Using the definition of the mean-Berezin norm, we explore some related new inequalities For instance, if G ∈ O ( K ) , then 1 2 ∥ | G | 2 r + | G ∗ | 2 ( 1 − r ) ∥ ber ⩾ ∥ G ∥ M ρ , where 0 ⩽ r ⩽ 1 and ρ is a mean such that ρ ⩽ ∇ .

The Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℍ = ℍ(Θ) over some (non-empty) set Θ, is defined by Ã(λ) = ⟨Ak̂λ,k̂λ ⟩ (λ ∈ Θ), where k ^ λ = k λ ‖ k λ ‖ is the normalized reproducing kernel of ℍ. The Berezin number of an operator A is defined by ber ( A ) = sup λ ∈ Θ | A ˜ ( λ ) | = sup λ ∈ Θ | 〈 A k ^ λ , k ^ λ 〉 | . In this paper, by using the definition of g-generalized Euclidean Berezin number, we obtain some possible relations and inequalities. It is shown, among other inequalities, that if Ai ∈ 𝕃(ℍ(Θ)) (i = 1,..., n), then ber g ( A 1 , … , A n ) ≤ g − 1 ( ∑ i = 1 n g ( ber ( A i ) ) ) ≤ ∑ i = 1 n ber ( A i ) in which g : [0, ∞) → [0, ∞) is a continuous increasing convex function such that g(0) = 0.

A functional Hilbert space is the Hilbert space of complex-valued functions on some set Θ ⊆ C that the evaluation functionals φ λ ( f ) = f ( λ ) , λ ∈ Θ are continuous on H . Then, by the Riesz representation theorem, there is a unique element k λ ∈ H such that f ( λ ) = 〈 f , k λ 〉 for all f ∈ H and every λ ∈ Θ . The function k on Θ × Θ defined by k ( z , λ ) = k λ ( z ) is called the reproducing kernel of H . In this study, we defined the weighted Davis-Wielandt Berezin number, and then we obtained some related inequalities. It is shown, among other inequalities, that if X ∈ L ( H ) and ν ∈ [ 0 , 1 ] , then 1 2 ( ber 2 ( X ν + | X ν | 2 ) + c ber 2 ( X ν − | X ν | 2 ) ) ≤ d w ber ν 2 ( X ) ≤ 1 2 ( ber 2 ( X ν + | X ν | 2 ) + ber 2 ( X ν − | X ν | 2 ) ) , where X ν = ( 1 − 2 ν ) X ∗ + X . Some bounds for the weighted Davis-Wielandt Berezin number are also established.

We present new bounds for the Berezin number inequalities which improve on the existing bounds. We also obtain bounds for the Berezin norm of operators as well as the sum of two operators.

In this paper, we find new upper bounds for the Berezin number of the product of bounded linear operators defined on reproducing kernel Hilbert spaces. We also obtain some interesting upper bounds concerning one operator, the upper bounds obtained here refine the existing ones. Further, we develop new lower bounds for the Berezin number concerning one operator by using their Cartesian decomposition. In particular, we prove that ber ( A ) ≥ 1 2 ber ( ℜ ( A ) ± J ( A ) ) , where ber(A) is the Berezin number of the bounded linear operator A.

In this paper, we present several Berezin number inequalities involving extensions of Euclidean Berezin number for n operators. Among other inequalities for (T₁, . . . , Tn ) ∈ 𝔹(𝓗) we show that ber p p ( T 1 , … , T n ) ≤ 1 2 p ber ( ∑ i = 1 n ( | T i | + | T i * | ) p ) , where p > 1.

In this paper, we generalize and refine some Berezin number inequalities for Hilbert space operators. Namely, we refine the Hermite-Hadamard inequality and some other recent results by using the concept of superquadraticity and convexity. Then we extend these inequalities for the Berezin number. Among other inequalities, it is shown that if S, T ∈ 𝓛(𝓗(Ω)) such that ber(T) ≤ ber(|S|) and f is a non-negative superquadratic function, then f (ber (T)) ≤ ber(f (|S|)) − ℓber ϵ f (∥S| − ber (T)|).

We prove some new Jensen type inequalities for the Berezin symbol of self-adjoint operators and some class of positive operators on the reproducing kernel Hilbert space ℋ(Ω) over some set Ω. Recall that the Berezin symbol A ~ of operator A on ℋ(Ω) is defined by the following special type of quadratic form: A ~ ( λ ) : = 〈 A k ^ λ , k ^ λ 〉 , λ ∈ Ω , where 𝑘𝜆 is the reproducing kernel of the space ℋ(Ω), i.e., f ( λ ) = 〈 f , k ^ λ 〉 for all 𝑓 ∈ ℋ(Ω) and λ ∈ Ω ; k ^ λ : = k λ ‖ k λ ‖ ℋ ( Ω ) is the normalized reproducing kernel.

The Berezin symbol à of an operator A on the reproducing kernel Hilbert space 𝓗(Ω) over some set Ω with the reproducing kernel kξ is defined by A ˜ ( ξ ) = 〈 A k ξ ‖ k ξ ‖ , k ξ ‖ k ξ ‖ 〉 , ξ ∈ Ω . The Berezin number of an operator A is defined by ber ( A ) : = sup ξ ∈ Ω | A ˜ ( ξ ) | . We study some problems of operator theory by using this bounded function Ã, including treatments of inner product inequalities via convex functions for the Berezin numbers of some operators. We also establish some inequalities involving of the Berezin inequalities.

In this paper, by the definition of Berezin number, we present some inequalities involving the operator geometric mean. For instance, it is shown that if X,Y,Z∈L(H) such that X and Y are positive operators, then berr((X♯Y)Z)≤ber((Z⋆YZ)rq2q+Xrp2p)−1pinfλ∈Ω([X˜(λ)]rp4−[(Z⋆YZ)˜(λ)]rq4)2, in which X♯Y=X12(X−12YX−12)12X12, p≥q>1 such that r≥2q and 1p+1q=1.