State carefully Rouché's theorem. Use it to show that the function $z^{4}+3+e^{i z}$ has exactly one zero $z=z_{0}$ in the quadrant

$$\{z \in \mathbb{C} \mid 0<\arg (z)<\pi / 2\}$$

and that $\left|z_{0}\right| \leqslant \sqrt{2}$.