The article is 'Hadamard and Calabi Yau conjectures on negatively curved an minimal surfaces' Nadirashvili. In the proof of proposition 4.3 it asserts that the function y is holomorphic. I'm not sure about it (actually i think it is false). What do you think about it? Thank you for your help.
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I agree, this part is wrong as stated, although the ideas behind it are sound. Pascal Collin and Harold Rosenberg did a great service to the community by publishing the note
in which they clarify and correct various details. In particular, the appendix of this note contains a cleaned-up proof of Proposition 3.4, which provides the missing details concerning the zeroes of the various functions involved.