On power deformations of univalent functions

TL;DR

This paper investigates the conditions under which power deformations of certain univalent functions remain univalent, revealing that the set of such parameters is linked to the variability region of a specific function related to the original function.

Contribution

It characterizes the parameter values for power deformations that preserve univalence and connects these to the variability region of $zf'(z)/f(z)$ for the class.

Findings

Identifies the set of $c$ for which $f_c$ is univalent.

Shows the set is described by the variability region of $zf'(z)/f(z).

Proves boundedness of strongly spirallike functions.

Abstract

For an analytic function $f(z)$ on the unit disk $|z|<1$ with $f(0)=f'(0)-1=0$ and $f(z)\ne0, 0<|z|<1,$ 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 $f\mapsto f_c$ 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.