convexity and complex square root

Question

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?

Answer 1

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 $\frac 12 (z_1 + z_2) = \frac 12$.