On power deformations of univalent functions

For an analytic function f (z) on the unit disk {pipe}z{pipe} < 1 with f (0) = f′(0) - 1 = 0 and f (z) ≠ 0, 0 < {pipe}z{pipe} < 1, we consider the power deformation f _c(z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator → f _c maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), {pipe}z{pipe} < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.