The Berezin range of a bounded operator A acting on a reproducing kernel Hilbert space \(\mathscr {H}\) is the set \({\textbf {Ber}}(A):= \{ \langle A\hat{k}_{\tau },\hat{k}_{\tau }\rangle : \tau \in \Theta \}\) , where \(\hat{k}_{\tau }\) is the normalized reproducing kernel for \(\mathscr {H}\) at \(\tau \in \Theta \) . The Berezin radius (number) and the Berezin norm of an operator A are defined by \({\textbf {ber}}\left( A\right) :=\underset{\tau \in \Theta }{\sup }\big \vert \langle {A}\hat{k}_{\tau },\hat{k}_{\tau }\rangle \big \vert \) and \(\left\| A\right\| _{{\textbf {ber}}}:=\underset{\tau ,\mu \in \Theta }{\sup }\big \vert \langle {A}\hat{k}_{\tau },\hat{k}_{\mu }\rangle \big \vert \) , respectively. In this paper, we obtain some estimations for the Berezin radius and the Berezin norm. It is shown, among other inequalities, that if \(A,B\in {{\mathscr {L}}}({{\mathscr {H}}})\) and \(1\le r\le 2\) , then \(\begin{aligned} {\textbf {ber}}^r(A^*B)&\le \frac{1}{2^{r\mu } p} {\textbf {ber}}^{r(1-\mu )}(A^*B)\left\| \vert A\vert ^{2}+ \vert B\vert ^{2}\right\| _{\textbf {ber}}^{r\mu }\\&\quad +\frac{1}{2^{r(1-\nu )}q} {\textbf {ber}}^{r\nu }(A^*B) \left\| \vert A\vert ^{2}+ \vert B\vert ^{2}\right\| _{\textbf {ber}}^{r(1-\nu )}\\ &\le \frac{1}{2} \left\| \vert A\vert ^{2r}+ \vert B\vert ^{2r}\right\| _{\textbf {ber}} \end{aligned}\) for all \(\mu , \nu \in [0,1]\) and \(p,q>0\) with \(\frac{1}{p}+\frac{1}{q}=1\) .
