Complex analysis (math): question about derivative

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je suis un expert en mathématique et j'ai résolus beaucoup de problèmes difficile que même mes profs trouvent parfois des difficultés a résoudre et j'ai une très bonne rédaction et les idées que j’utilise sont très claire que tout le monde peut comprendre . maintenant je connais la réponse je n'entends que votre accepte . suppose on the contrary that the maximum value of IfI is taken out of the boundary cercle {z /z<=1/2} , so there existe z / IzI <1/2 and because {z/IzI<1/2} is an open so f '(z)=0 which contradict our assumption. the result is not true if we omitt the assumption , we can show that by taking the function f(z) = k where k is a complex constant