Complete refinements of the Berezin number inequalities

Abstract: In this paper, several refinements of the Berezin number inequalities are obtained. We generalize inequalities involving powers of the Berezin number for product of two operators acting on a reproducing kernel Hilbert space $\mathcal H=\mathcal H(\Omega)$ and also improve them. Among other inequalities, it is shown that if $A,B\in {\mathcal B}(\mathcal H)$ such that $|A|B=B^{*}|A|$, $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ satisfying $f(t)g(t)=t\,(t\geq 0)$, then \begin{align*} &\textbf{ber}^{p}(AB)\leq r^{p}(B)\times\&\left(\textbf{ber} \big(\frac{1}{\alpha}f^{\alpha p}(|A|)+\frac{1}{\beta}g^{\beta p}(|A^{*}|)\big)-r_{0}\big(\langle f^{2}(|A|)\hat{k}{\lambda},\hat{k}{\lambda}\rangle^{\alpha p/4} -\langle g^{2}(|A^{*}|)\hat{k}{\lambda},\hat{k}{\lambda}\rangle^{\beta p/4}\big)^{2}\right) \end{align*} for every $p\geq 1, \alpha\geq\beta>1$ with $\frac{1}{\alpha}+\frac{1}{\beta}=1$, $\beta p\geq2$ and $r_{0}=\min{\frac{1}{\alpha},\frac{1}{\beta}}$.