Some extensions of Berezin number inequalities on operators

In this paper, we establish some upper bounds for Berezin number inequalities including of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if $T=\left[\begin{array}{ccc}0& X,Y& 0\end{array}\right]$, then \begin{align*} $\mathbf{\text{ber}}$^{r}(T)\leq 2^{r-2}\left($\mathbf{\text{ber}}$(f^{2r}(|X|)+g^{2r}(|Y^|))+$\mathbf{\text{ber}}$(f^{2r}(|Y|)+g^{2r}(|X^|))\right)\ -2^{r-2} \inf_{|(k_{λ_{1}},k_{λ_{2}})|=1} η(k_{λ_{1}},k_{λ_{2}}), \end{align*} where $\eta (k_{{\lambda }_1},k_{{\lambda }_2})={(⟨(f^{2r}(\mathrm{\mid }X\mathrm{\mid })+g^{2r}(\mathrm{\mid }{Y}^{\ast }\mathrm{\mid }))k_{{\lambda }_2},k_{{\lambda }_2}⟩}^{\frac{1}{2}}-{⟨(f^{2r}(\mathrm{\mid }Y\mathrm{\mid })+g^{2r}(\mathrm{\mid }{X}^{\ast }\mathrm{\mid }))k_{{\lambda }_1},k_{{\lambda }_1}⟩}^{\frac{1}{2}}{)}^2$, $X,Y$ are bounded linear operators on a Hilbert space $\mathcal{H}=\mathcal{H}(\mathrm{\Omega })$, $r\ge 1$ and $f$, $g$ are nonnegative continuous functions on $[0,\mathrm{\infty })$ satisfying the relation $f(t)g(t)=t(t\in [0,\mathrm{\infty }))$.