Geodesic
On the difference $f(B)-f(A)$ for unbounded self-adjoint operators in the perturbation theory
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 213-221

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The main result of the paper is the estimate $$ \|f(B)-f(A)\|\le c\biggl[\log\biggl(1+\frac1{\|B-A\|}\biggr)+7\biggr]^2\|B-A\|, $$ obtained for Lipschitz functions, with some conditions of the growth of the functions at infinity. 
@article{ZNSL_1982_107_a17,
     author = {J. B. Farforovskaja},
     title = {On the difference $f(B)-f(A)$ for unbounded self-adjoint operators in the perturbation theory},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {213--221},
     publisher = {mathdoc},
     volume = {107},
     year = {1982},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_107_a17/}
}
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J. B. Farforovskaja. On the difference $f(B)-f(A)$ for unbounded self-adjoint operators in the perturbation theory. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part X, Tome 107 (1982), pp. 213-221. http://geodesic.mathdoc.fr/item/ZNSL_1982_107_a17/