Stability of the Low pass filter after Spectral Transformation
The filter is stable if the all the poles of the transformed filter, i.e., filter obtained after applying spectral transformation, lie inside the unit circle. For example, if $H(z)$ is the original filter and $F(z)$ is the filter response of the transformation filter, then the spectrally transformed filter response is given as
\begin{align}
G(z) = H(F(z)) = H(z)|_{z^{-1}\leftarrow 1/F(z)}
\end{align}
where
\begin{align}
\frac{1}{F(z)} = \frac{1-\beta z}{z - \beta}.
\end{align}

Check if the poles of $G(z)$ lie inside the unit circle. If yes, then $G(z)$ is stable. Note that $F(z)$ is an all-pass filter. Therefore, the regions of stability and instability of $H(z)$ are preserved in $G(z)$.