9. Inequality: $\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9$ has the solution set

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9. Inequality: $\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9$ has the solution set
10. $\left[\pi-\arctan \frac{4}{3}, \pi\right)$

Since $|z-i|=1$, the locus of point $Z$ is a circle with center at $(0,1)$ and radius 1 (excluding points $(0,0)$ and $(0,2)$).

Therefore, $\frac{z}{z-2 i}$ is a pure imaginary number, so $\frac{\omega-2 i}{\omega}$ is also a pure imaginary number. The locus of point $\omega$ should be a circle with center at $(0,1)$ and radius 1 (excluding points $(0,0)$ and $(0,2)$). Thus, $\arg (\omega-2) \in \left[\pi-\arctan \frac{4}{3}, \pi\right)$.