Abstract. We give operator analogues of some classical inequalities, including Hardy and HardyHilbert type inequalities for numbers. We apply these operator forms of such inequalities for proving some power inequalities for the so-called Berezin number of self-adjoint and positive operators acting on Reproducing Kernel Hilbert Spaces (RKHSs). More precisely, we prove thatfor some constants C > 1. We also use reproducing kernels technique to estimate dist (A,U ) , where U is the set of all unitary operators on a RKHS H = H (Ω) over some set Ω, for some operator A on H (Ω) .Mathematics subject classification (2010): 47A63.