Abstract
The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano’s ones for real functions on an interval is deduced in a very simple way from Cauchy’s theorem for holomorphic functions. A more complicated proof, using Cauchy’s argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic map**s from ℂn to ℂn are obtained using Brouwer degree.
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To Haïm Brezis, with friendship and admiration
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Mawhin, J. Bolzano’s theorems for holomorphic map**s. Chin. Ann. Math. Ser. B 38, 563–578 (2017). https://doi.org/10.1007/s11401-017-1083-8
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DOI: https://doi.org/10.1007/s11401-017-1083-8
Keywords
- Holomorphic function
- Hadamard-Shih’s conditions
- Poincaré-Miranda’s conditions
- Bolzano’s theorem
- Cauchy’s theorem
- Brouwer fixed point theorem
- Brouwer degree