Let $P_z(t)=\frac{1}{\pi}Im(\frac{1}{t-z})=\frac{1}{\pi} \frac{y}{(x-t)^2+y^2}$
where $0<y=Im(z)$ and $x=Re(z)$
$P$ is the Poisson kernel for the upper half plane.
If $t \in \Bbb{R}$ how can we derive that $P_z(t) \leq \frac{c_z}{1+t^2}$ where $c_z$ is a constant depending on $z$?
Thank you in advance for your help.
Best Answer
Certainly. Since your bound can depend on $z$ all that you are asking is if $\frac {1+t^{2}} {(x-t)^{2}+y^{2}}$ is bounded in $t$. This is a continuous function which tends to $1$ as $|t| \to \infty$ so it is bounded.