Is it true that if a holomorphic function in the unit disk converges uniformly to the 0 function some connected arc of the unit circle, this function is globally null?If that is true, this would stand for a uniqueness fact and so how to recover some holomorphic function from its boundary value on an arc (if the uniform convergence still holds on the arc)

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Jimena Torres · Accepted Answer

Let the (open) arc on which the boundary values of   f vanish be   A. Since the boundary values of   f on   A are real, by the Schwarz reflection principle we know that the function  g  (  z  )  =      {                                                f          (          z          )                          ,          |          z          |          &amp;lt;          1                                                0                          ,          z          ∈          A                                                                f              (              1                              /                                            z                ¯                            )                        ¯                                    ,          |          z          |          &amp;gt;          1                        is holomorphic on the connected open set       D    ∪  A  ∪  (      C    ∖            D        ¯    ). Since   g vanishes on a non-discrete set, the identity theorem yields   g  ≡  0, in particular   f  ≡  0 follows.

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