How can I show that $z^4-2z+3$ has no zeros within the unit circle in the complex plane?

Question

How can I show that $z^4-2z+3$, $z \in \mathbb{C}$, has no zeros within the unit circle in the complex plane? It looks like the Rouche theorem, but i still cannot do it. Please help. Thanks in advance.

Answer 1

We have $$|z^4-2z+3|\ge 3-2|z|-|z|^4.$$ Hence the equation does not admit solutions for $|z|<1.$ In the case $|z|=1$ the equality $z^4-2z=-3$ implies $z^4=-1$ and $z=1,$ which leads to a contradiction.

Answer 2

A proof using Rouché's theorem.

Consider $f(z):=z^4+3$, $g(z):=−2z$.

On the disk $D_{\varepsilon}$ with center 0 and radius $1−\varepsilon$. On $\partial D_{\varepsilon}$,

$$|f(z)| \ge 2+3 \varepsilon \ \ \text{and} \ \ |g(z)| \le 2(1−\varepsilon)$$

for $\varepsilon$ sufficiently small, allowing Rouché's theorem to be applied, proving that there is no zero inside $D_{\varepsilon}$.

(thanks to Lutz Lehmann who has pointed an error of mine, now fixed).