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Determine the number of points $z$ in the unit disk such that $e^z = 3z^4$.

I want to use the Rouche's Theorem, but I could not construct such two proper functions.

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Denote $f(z)=e^z-3z^4$, $g(z)=3z^4$,then $$|f(z)-g(z)|=|e^z|\leqslant e^{|z|}=e<3=3|z^4|=|g(z)|$$whenever $|z|=1$, i.e. $z\in\mathbb{D}$.Then use the Rouche's Theorem.

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