Example 9. Find the number of roots of the equation

$$
z^{2}-a e^{z}=0, \quad \text { where } \quad 0<a<e^{-1}
$$

in the unit circle $|z|<1$.
Solution. Let $f(z)=z^{2}$ and $\varphi(z)=-a e^{z}$. On the circle $|z|=1$ we have

$$
\begin{aligned}
& |f(z)|=\left|z^{2}\right|=1 \\
& |\varphi(z)|=\left|-a e^{z}\right|=a\left|e^{z}\right|=a\left|e^{x+i y}\right|=a e^{x} \leqslant a e|\varphi(z)|$, if $|z|=1$. The function $f(z)=z^{2}$ in the circle $|z|0, \quad \text { since } \quad 00, \quad \text { since } \quad a1), \quad|z|\frac{e^{R}}{R^{n}}, \quad|z|<R$.
452. $z^{2}-\cos z=0, \quad|z|<2$.
453. $z^{4}-\sin z=0, \quad|z|<\pi$.
454. $z^{2}+\operatorname{ch} i z=0, \quad|z|<0.5$.
455. $\operatorname{ch} z=z^{2}-4 z,|z|<1$.
456. $2^{z}=4 z, \quad|z|<1$.

Note: There are some parts in the original text that seem to be incomplete or contain errors, such as the conditions and inequalities. Please review the original text for clarity.