An extension of the Euclidean Berezin number

Abstract

The Berezin transform $\widetilde{A}$ of an operator $A$, acting on the
reproducing kernel Hilbert space ${\mathscr H}={\mathscr H(}\Theta )$ over
some (non-empty) set $\Theta $, is defined by $\widetilde{A}(\lambda
)=\langle A\hat{k}_{\lambda },\hat{k}_{\lambda }\rangle \,\,\,(\lambda \in
\Theta )$, where $\hat{k}_{\lambda }=\frac{{k}_{\lambda }}{\Vert {k}%
_{\lambda }\Vert }$ is the normalized reproducing kernel of ${\mathscr H}$.
The Berezin number of an operator $S$ is defined by $\mathbf{ber}(A)=%
\underset{\lambda \in
%TCIMACRO{\U{211d} }%
%BeginExpansion
{\Theta}
%EndExpansion
}{\sup }\big\vert\widetilde{A}(\lambda )\big\vert=\underset{\lambda \in
%TCIMACRO{\U{211d} }%
%BeginExpansion
{\Theta}
%EndExpansion
}{\sup }\big\vert\langle A\hat{k}_{\lambda },\hat{k}_{\lambda }\rangle \big\vert$.
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 $A_{i}\in \mathscr{L(H}(\Theta))\,\,(i=1,\ldots,n)$, then
\begin{align*}
\textbf{ ber}_{g}(A_{1},...,A_{n})\leq
g^{-1}\left(\sum_{i=1}^{n}g\left(\textbf{ ber}(A_i)\right)\right)
\leq\sum_{i=1}^{n}\textbf{ ber}(A_i),
\end{align*}
in which $g:[0,\infty)\rightarrow [0,\infty)$ is a continuous increasing convex function such that $g(0)=0$.