Geodesic
Effective estimates for derivatives of inverse functions
Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 761-791

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We use the ‘graph’ method to obtain estimates for the derivatives of any order of inverse functions in terms of those of the original functions. We construct explicit asymptotics of the estimates obtained as the order of the derivative tends to infinity. For analytic functions and functions in Gevrey's class, we obtain explicit estimates for all derivatives of the inverse functions. 
@article{IM2_2008_72_4_a6,
     author = {Yu. V. Lysenko},
     title = {Effective estimates for derivatives of inverse functions},
     journal = {Izvestiya. Mathematics },
     pages = {761--791},
     publisher = {mathdoc},
     volume = {72},
     number = {4},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a6/}
}
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Yu. V. Lysenko. Effective estimates for derivatives of inverse functions. Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 761-791. http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a6/