This is a continuation of our papers [3] and [4]. In those papers we obtained estimates for finite differences (δ_kf) (A) = f(A + K) -f(A) of the order 1 and mathematcal equation represented of the order m for certain classes of functions f , where A and K are bounded self-adjoint operators. In this paper we extend results of [3] and [4] to the case of unbounded self-adjoint operators A. Moreover, we obtain operator Bernstein type inequalities for entire functions of exponential type. This allows us to obtain alternative proofs of the main results of [3]. We also obtain operator Bernstein type inequalities for functions of unitary operators. Some results of this paper as well as of the papers [3] and [4] were announced in [2].