Suppose $a,b \in \Bbb C$, and denote by $B(a,\beta)$ the Euclidean disc centered at $a$ with radius $\beta$. Here $dA$ will represent the usual Lebesgue area measure.
Can I say there exists a constant $C_{\beta} > 0$ depending only on $\beta$ such that
$$\int_{B(a,\beta)} e^\frac{-|z|^2}{2}dA(z) \leq C_{\beta} \int_{B(b,\beta)} e^\frac{-|z|^2}{2}dA(z)$$
if the Euclidean distance $|a-b| < \delta $ for some fixed $\delta > 0$ independent of both $a$ and $b$?
I believe this is trivially true if $|a| \geq |b|$ since $e^\frac{-|z|^2}{2}$ decreases strictly as $|z|$ increases, but how about if $|b| > |a|$?