On power deformations of univalent functions

Abstract

Abstract. For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f ′ (0)−1 = 0 and f (z) = 0, 0 < |z| < 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 → 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), |z| < 1, for the class in most cases which we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.

An Extremal Problem for Univalent Functions

“…See [6] and [7] for details about the power deformation. We now have j f c ðzÞj ¼ jzj expða log j f ðzÞ=zj À b arg½ f ðzÞ=zÞ:…”