Improvements of Berezin number inequalities

Abstract

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*} \textbf{ber}^{r}(H_α(A,B))&\leq\frac{\|X\|^{r}}{2}\textbf{ber}(A^{r}+B^{r})&\leq\frac{\|X\|^{r}}{2}\textbf{ber}(αA^{r}+(1-α)B^{r})+\textbf{ber}((1-α)A^{r}+αB^{r}), \end{align*} where $H_{\alpha }(A,B)=\frac{A^{\alpha }X{B}^{1-\alpha }+A^{1-\alpha }X{B}^{\alpha }}{2}$, $0\le \alpha \le 1$ and $r\ge 2$.