I'm trying to show that the expression
$\omega_i^2 \mid \omega \mid^{-\beta}$
(where $\mid \omega \mid = ( \omega_1^2 + \omega_2^2)^{\frac{1}{2}}$ and $i = [1, 2]$) is defined at the origin under certain conditions. Here's my attempt. I'd like to know if its correct.
Because $\mid \omega \mid^2 = \omega_1^2 + \omega_2^2 \Rightarrow \omega_i^2 \le \mid \omega \mid^2$ for any $i$. Therefore, $\omega_i^2 \mid \omega \mid^{-\beta} \le \mid \omega \mid^2 \mid \omega \mid^{-\beta} = \mid \omega \mid^{2 - \beta}$.
Therefore the expression is defined at the origin if $2 - \beta > 0$ or $\beta < 2$.
Edit: in addition can it be shown that because $\omega_i \le \mid \omega \mid$ (I assume this is obvious), $\omega_i^3 \le \mid \omega \mid^3$ and by combining both inequalities it is possible to show in general that $\omega_i^n \le \mid \omega \mid^n$ for odd and even values of $n$.