Suppose I have two points in $\mathbb{R}^2$,
$$ x = (x_1,x_2) = r_x(\cos\theta_x,\sin\theta_x)\quad\text{ and }\quad y = (y_1, y_2) = r_y(\cos\theta_y,\sin\theta_y), $$
and the angle coordinate is to lie inside $(-\pi,\pi)$. Consider 'a variant' of the complex square root map defined as
$$ \omega(x) = \sqrt{r_x}\left(\cos\frac{\theta_x}{2},\sin\frac{\theta_x}{2}\right). $$ This maps $\mathbb{R}^2\setminus\{(x,0) \,|\, x\leq 0\}$ onto the half-space with positive $x$ component.
I'm curious what is the general relation between
$$ |x-y|\quad\text{ and }\quad |\omega(x) - \omega(y)|. $$ Since $\omega$ is a square-root map, the naive intuition tells me that
$$ |\omega(x) - \omega(y)| \sim |x-y|^{1/2}, $$ but a closer look seems to reveal that the situation is more nuanced. For instance, , if we let $r_x = r_y = k$, then a tedious calculation revals that
$\frac{|\omega(x) - \omega(y)|}{|x-y|^{1/2}} = \frac{|\omega(x) - \omega(y)|^4}{|x-y|^{2}} = \tan\left(\frac{\theta_x - \theta_y}{4}\right)^2$
and the RHS goes all the way from 0 to $+\infty$ for admissible values of the angles. In particular there is no dependence on $k$.
The particular aspect I'm interested in can be summarised as follows. What is the largest $\lambda$ for which there exists $C > 0$ such that
$$ |\omega(x) - \omega(y)| \geq C|x-y|^{\lambda}, $$ for all $x,y\in \mathbb{R}^2\setminus\{(x,0) \,|\, x\leq 0\}$. Is it $\lambda = \frac{1}{2}$? Can it even be stated in this way?
Many thanks for any potential insight!