Improved inequalities for the Berezin number

A functional Hilbert space is the Hilbert space of complex-valued functions on some set C that the evaluation functionals ( f ) = f () , are continuous on H .The Berezin number(radius) of an operator T is defined by ber(T ) = sup T k , k , where the operator T acts on the reproducing kernel Hilbert space H = H () over some(non-empty) set .In this paper, we give some Berezin number inequalities.Moreover, we present some inequalities involving the weighted Berezin number of operators on the reproducing kernel Hilbert space.