Geodesic
Estimate of $\|f(A_1, A_2)-f(B_1, B_2)\|$ for a~couple of selfadjoint commuting operators.
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 175-177

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The estimate $\|f(A_1, A_2)-f(B_1, B_2)\|\leqslant C(\log K+4)^3(\|A_1-B_1\|+\|A_2-B_2\|)$ is announced. The proof is outlined. 
@article{ZNSL_1984_135_a14,
     author = {J. B. Farforovskaja},
     title = {Estimate of $\|f(A_1, A_2)-f(B_1, B_2)\|$ for a~couple of selfadjoint commuting operators.},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {175--177},
     publisher = {mathdoc},
     volume = {135},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_135_a14/}
}
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J. B. Farforovskaja. Estimate of $\|f(A_1, A_2)-f(B_1, B_2)\|$ for a~couple of selfadjoint commuting operators.. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 175-177. http://geodesic.mathdoc.fr/item/ZNSL_1984_135_a14/