# convexity and complex square root

I have a convex hull enumerated by vertices $${a_1,\cdots, a_n}$$ in the complex plane. I am performing the (complex)square root operation over these vertices. How would the image look like after this operation? whether the convexity is still preserved? Does it going to make a difference if we consider only the principal square root?

• If you start with just one point $-1$ then the convex hull of this is again $\{-1\}$. The square roots of the points in this set consist of two points $i$ and $-i$ so we get a non-convex set. – Kavi Rama Murthy Aug 14 at 5:29

A counterexample: The image of the segment $$[-i, i]$$ under the (principle value of the) square root is not convex.
It contains the points $$z_{1, 2} = \sqrt{\pm i} = \frac{1 \pm i}{2}$$, but not the point $$\frac 12 (z_1 + z_2) = \frac 12$$.