New estimations for the Berezin number inequality

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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X, Y, Z\in {\mathcal{L}}(\mathcal{H})$\end{document} 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,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \operatorname{ber}^{r} \bigl( ( X\mathbin{\sharp} Y ) Z \bigr) &\leq \operatorname{ber} \biggl(\frac{ ( Z^{\star }YZ ) ^{\frac{rq}{2}}}{q}+ \frac{X^{ \frac{rp}{2}}}{p} \biggr) -\frac{1}{p}\inf_{\lambda \in \varOmega } \bigl( \bigl[ \widetilde{X} ( \lambda ) \bigr] ^{\frac{rp}{4}}- \bigl[ \widetilde{ \bigl( Z^{\star }YZ \bigr) } ( \lambda ) \bigr] ^{ \frac{rq}{4}} \bigr) ^{2}, \end{aligned}$$ \end{document} in which X♯Y=X12(X−12YX−12)12X12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X\mathbin{\sharp} Y=X^{\frac{1}{2}} ( X^{-\frac{1}{2}}YX^{- \frac{1}{2}} ) ^{\frac{1}{2}}X^{\frac{1}{2}}$\end{document}, p≥q>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\geq q>1$\end{document} such that r≥2q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$r\geq \frac{2}{q}$\end{document} and 1p+1q=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{1}{p}+\frac{1}{q}=1$\end{document}.


Introduction and preliminaries
We denote the C * -algebra of all bounded linear operators on a separable complex Hilbert space H with L(H). An operator X ∈ L(H) is called positive if Xx, x ≥ 0 for every x ∈ H, and in this case we write X ≥ 0. The numerical range and numerical radius of X ∈ L(H) are respectively defined by W (X) := { Xf , f : f ∈ H, f = 1} and w(X) := sup{|f | : f ∈ W (X)}. We denote by F(Ω) the set of all complex-valued functions on a nonempty set Ω. Let H = H(Ω) ⊂ F(Ω) be a Hilbert space. The Riesz representation theorem makes certain that a functional Hilbert space has a reproducing kernel, which is a function k λ : Ω × Ω → H, that is called the reproducing kernel enjoying the reproducing property k λ := k(·, λ) ∈ H (λ ∈ Ω) such that f (λ) = f , k λ H , in which λ ∈ Ω and f ∈ H (see [18]). For {ξ n (z)} n≥0 , an orthonormal basis of the space H(Ω), the reproducing kernel can be presented as follows: where k λ = k λ k λ is the normalized reproducing kernel of H (see [7]). Karaev in [13][14][15] defined the Berezin set and the Berezin number for operator X as follows: respectively. Moreover, the Berezin number of two operators X, Y satisfies the following properties: Also, we know that for all X ∈ L(H). In some recent papers, several Berezin number inequalities have been investigated by authors [3-6, 9, 10, 12, 21, 22]. Assume that X 1 , . . . , X n ∈ L(H) and p ≥ 1. In [3], the generalized Euclidean Berezin number of X 1 , . . . , X n is defined as follows: If p, q > 1 with 1 p + 1 q = 1, then the Young inequality is the inequality where x and y are positive real numbers (see [11]). A refinement of (1) was obtained by Kittaneh and Manasrah [17] xy where r 0 = min{ 1 p , 1 q } or equivalently in which ν ∈ [0, 1] and r 0 = min{ν, 1 -ν}. For positive operators X, Y ∈ L(H), the operator geometric mean is the positive opera- where it has the property X Y = Y X. A matrix mean inequality was established by Bhatia and Kittaneh in [8], and later this inequality was generalized in [18]. A matrix Young inequality was obtained by Ando in [1]. The matrix mean inequality and the matrix Young inequality were considered with the numerical radius norm by Salemi and Sheikhhosseini in [19,20].
In this paper, we get some upper bounds for the Berezin number of the (X Y )Z on reproducing kernel Hilbert spaces (RKHS), where Z ∈ L(H) is arbitrary, and give some Berezin number inequalities. We also present some inequalities for the generalized Euclidean Berezin number.

Main results
We need the following lemma to prove our results (see [16]).
Proof Using the Cauchy-Schwarz inequality, we get for all λ ∈ Ω. By using the Young inequality and (2), we get and it follows from inequality (4) that for all λ ∈ Ω. This implies that Taking the Z = I in inequality (5), we have the following result.

Corollary 3 Let X, Y ∈ L(H) be positive operators, and let p
for all r ≥ 2 q .
Proof By inequality (2), we have for all λ, μ ∈ Ω and taking supremum over λ, μ ∈ Ω in the above inequality, we get X ber Z YZ ber In the next theorem we show an upper bound for the generalized Euclidean Berezin number.