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Thank you, Karl Schwede. I won't write anything related, because my research field is very far from this. I just came up with this demonstration while "playing" with some different problems on a Sunday. :)
Thank you, Alexandre Eremenko. No, it doesn't give any new insight, that's why I wondered in the first place if it was worth publishing it. Then, I thought: why not? It's still interesting at least as a curiosity.
thank you! I'll submit to the American Mathematical Monthly as you suggest, and let's see how it goes. (The theorem is very common, and I won't certainly write any book on the subject.)
@AnthonyQuas, no, no advantages whatsoever. It's just interesting because it is more simple and quick than the standard demonstration, although it relies on more advanced math.