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The Berezin symbol $\tilde{A}$ of an operator $A$ on the reproducing kernel Hilbert space $\mathcal{H}\left( \Omega\right) $ over some set $\Omega$ with the reproducing kernel $k_{\lambda}$ is defined by $$ \tilde{A}(\lambda)=\left\langle {A\frac{{k_{\lambda}}}{{\left\Vert {k_{\lambda}}\right\Vert }},\frac{{k_{\lambda}}}{{\left\Vert {k_{\lambda} }\right\Vert }}}\right\rangle ,\ \lambda\in\Omega. $$ The Berezin number of an operator $A$ is defined by $$ ber(A):=\sup_{\lambda\in\Omega}\left\vert {\tilde{A}(\lambda)}\right\vert . $$ We study some problems of operator theory by using this bounded function $\tilde{A}$, including estimates for Berezin numbers of some operators, including truncated Toeplitz operators. We also prove an operator analog of some Young inequality and use it in proving of some inequalities for Berezin number of operators including the inequality $ber\left( {AB}\right) \leq ber\left( A\right) ber\left( B\right) ,$ for some operators $A$ and $B$ on $\mathcal{H}\left( \Omega\right) $. Moreover, we give in terms of the Berezin number a necessary condition for hyponormality of some operators.
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