When I took complex analysis from Ahlfors in 1970 or so (using his textbook), when we came to the section on this, he made a remark in class. For a long time it was thought this could not be proved without integration. Today proofs without integration are known. But they are much more difficult than the proof in the textbook with integration. He did not give any more explanation. Maybe (?) if there is a more recent edition of his book he added the remark there?