Show that if $a>e$, the equation $az^n=e^z$ admit $n$ roots in the unit disk - Rouché theorem

complex-analysis roots

Hint. On $\{\lvert z\rvert=1\}$ we have $$ \lvert\mathrm{e}^z\rvert\le \mathrm{e}<a\le \lvert az^n-\mathrm{e}^z\rvert+\lvert az^n\rvert. $$

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