It is not too hard to prove that if $z_1=r_1e^{it_1},z_2=r_2e^{it_2}$ with $|t_j|\leq \pi/2$ are two points in the unit disk $\mathbb D:=\{z\in \mathbb C: |z|<1\}$, not co-linear with $z=1$, then for every $z=re^{it}$ in the triangle $\Delta:=\langle z_1,z_2,1\rangle$ (which is defined to be the convex hull of these three points) one has $$\frac{|1-z|}{1-|z|}\leq \max\left\{\frac{|1-z_1|}{1-|z_1|},\frac{|1-z_2|}{1-|z_2|}\right\}.$$ Does the following formula also holds: $$\frac{|t|}{1-r}\leq \max\left\{ \frac{|t_1|}{1-r_1},\frac{|t_2|}{1-r_2} \right\}?$$
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