Convexity of the Berezin range of composition operators induced by automorphisms of the Unit Ball in ${\mathbb{C}}^n$

We study the convexity of the Bergman space Berezin range of composition operators induced by unit disk and unit ball automorphisms. We obtain several new results characterizing convexity of the Berezin range for linear fractional unit disk and unit ball automorphisms composed with unitary maps.