Estimate on expression involving elements in the unit disc in $\Bbb C$ Suppose $|\lambda|<1, Re\lambda < 0, Im\lambda > 0$. What is the largest lower bound independent of $\lambda$ on the expression $$Re\lambda - \frac{2\pi}{n+2}Im\lambda$$ where $n$ is a positive integer? If you like, stipulate that $n$ be as large as you need for the problem to make sense, though make it as small of a lower bound on $n$ as you can, if that makes sense.

Let $x = -\Re(\lambda), y = \Im(\lambda)$ and $c = \frac{2\pi}{n+2}$. The above problem is then equivalent to $$\begin{equation*} \begin{aligned} & {\text{minimize}} & & -x -cy\\\ & \text{subject to} & & x^2 + y^2 < 1 \\\ & & & x>0 \\\ & & & y>0 \end{aligned} \end{equation*}$$

This is a pretty tame convex program, and has a standard solution via the KKT conditions. In particular, the infimum is $-\sqrt{1+c^2}$, and the point $\frac{1}{\sqrt{1+c^2}} (-1,c)$ is the limit point of the region where this value is attained.