If I want to send $z \to \frac{z}{\sqrt{2}}$ in the following expression
$$ \int_{B(w, \beta)} e^{-|z|^2} dA(z)$$
where $B(w, \beta)$ is the Euclidean ball centered at $w$ with radius $\beta$ and $dA$ is the usual, normalized Lebesgue measure on $\mathbb C$, is there anyway to write the resulting expression as larger than some constant $C_{\beta}$ times
$$ \int_{B(w, \beta)} e^{\frac{-|z|^2}{2}} dA(z)?$$
I've never actually had to scale a variable in a transformation. Usually it's just a basic translation, which I can picture in my head more easily.
If you set $w = c z$ then $$ \int_{B(a,r)} e^{-|z|^2} \, dA(z) = \int_{B(ca,cr)} e^{-|w/c|^2} \, c^{-n} \, dA(w) = c^{-n} \int_{B(ca,cr)} e^{-|w/c|^2} \, dA(w) $$ where $n$ is the dimension of the space.