Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities

This is a continuation of our papers \cite{AP2} and \cite{AP3}. In those papers we obtained estimates for finite differences $(\D_Kf)(A)=f(A+K)-f(A)$ of the order 1 and $(\D_K^mf)(A)\df\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big)$ of the order $m$ for certain classes of functions $f$, where $A$ and $K$ are bounded self-adjoint operator. In this paper we extend results of \cite{AP2} and \cite{AP3} 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 \cite{AP2}. We also obtain operator Bernstein type inequalities for functions of unitary operators. Some results of this paper as well as of the papers \cite{AP2} and \cite{AP3} were announced in \cite{AP1}.