Find all zeros of the equation $e^z = 1+2z$ in the unit disk. I tried solving the pair of equations 
\begin{align}
e^x\cos y&amp;=1+2x\\
e^x\sin y&amp;= 2y
\end{align}
but I got stuck.
let $f(z) = e^{z}$ and $g(z) = -(1+2z)$ then on the boundary $e^{i\theta}$, 

$$ |f(z)| = |e^{\cos \theta}| &lt; \sqrt{5 + 4 \cos(\theta)} \tag{1}$$
Since $g(z)$ has  one root on $|z|&lt;1$, by Rouché's Theorem $f(z)+g(z)$ has one root on $|z|&lt; 1 $. In particular, $z=0$ satisfies the equation and it is only the root of it as well.

To show that $(1)$ holds, one can simply show that $h(x) = e^x - \sqrt{5 + 4x}$ is smooth function on $[1,1]$ has $h(x)$ has minimum value of $3-e$ on $x=1$.
Hint: one solution is quite easy.  It also happens to be the only one.