Improvements of Berezin number inequalities

In this paper, we generalize several Berezin number inequalities involving product of operators. For instance, we show that if $A,B$ are positive operators and $X$ is any operator, then \begin{align*} $\mathbf{\text{ber}}$^{r}(H_{\alpha}(A,B))&\leq\frac{|X|^{r}}{2}$\mathbf{\text{ber}}$(A^{r}+B^{r})&\leq\frac{|X|^{r}}{2}$\mathbf{\text{ber}}$(\alpha A^{r}+(1-\alpha)B^{r})+$\mathbf{\text{ber}}$((1-\alpha)A^{r}+\alpha B^{r}), \end{align*} where $H_{\alpha}(A,B)=\frac{A^\alpha XB^{1-\alpha}+A^{1-\alpha} XB^{\alpha}}{2}$,$0\leq\alpha\leq1$and$r\geq2$.