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