Geodesic
New proof of the uniqueness theorem for the differentiable complex-variable functions
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VI, Tome 40 (1974), pp. 119-126

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This proof is based on some results of [2] and on bounds for functions having many zeros inside a sufficiently small sphere. 
@article{ZNSL_1974_40_a13,
     author = {V. P. Orevkov},
     title = {New proof of the uniqueness theorem for the differentiable complex-variable functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {119--126},
     publisher = {mathdoc},
     volume = {40},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1974_40_a13/}
}
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V. P. Orevkov. New proof of the uniqueness theorem for the differentiable complex-variable functions. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part VI, Tome 40 (1974), pp. 119-126. http://geodesic.mathdoc.fr/item/ZNSL_1974_40_a13/