Complete refinements of the Berezin number inequalities

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 $r_{0}=\min\{\frac{1}{\alpha},\frac{1}{\beta}\}$.