The Berezin transform (A) over tilde and the Berezin number of an operator A on the reproducing kernel Hilbert space over some set Omega with normalized reproducing kernel (k) over cap (lambda) are defined, respectively, by (A) over tilde(lambda) = < A((k) over cap (lambda), (k) over cap (lambda)>, lambda is an element of Omega and ber(A) := sup(lambda is an element of Omega) |($) over tilde(lambda)|. A straightforward comparison between these characteristics yields the inequalities ber (A) <= 1/2 (||A||(ber) + ||A2||(1/2)(ber)). In this paper, we study further inequalities relating them. Namely, we obtained some refinements of Berezin number inequalities involving convex functions. In particular, for A is an element of B (H) and r >= 1 we show that