Given $p(z)=i z^{5}-8 z^{4}-\pi$,
How many zeros there is for $p(z)$ inside $ D_{1}(0) \cap\{z \mid \operatorname{Im}(z)>0\}$?
I can use Rouché theorem to infer how many zeros there are in the whole unit disk, but how do I infer the amount of zeros in the given domain?
Let $g(z) = 8z^4 + \pi$. Then on the unit disk we have $$ |iz^5| = 1 < 8 - \pi \leq |8z^4 + \pi| \leq |iz^5-8z^4-\pi| + |8z^4+\pi|. $$ Then, on the real axis in $(-1, 1)$ we have $$ |iz^5| < \pi \leq |8z^4 + \pi| \leq |iz^5-8z^4-\pi| + |8z^4+\pi|. $$ Since there are 2 zeros of $g$ in $\mathbb{D} \cap \mathbb{H}$, we know that there are 2 zeros of $p$ in $\mathbb{D} \cap \mathbb{H}$ aswell.