Where is $f(z)/f'(z)$ univalent?

TL;DR

This paper investigates the univalence of the operator $f(z)/f'(z)$ for functions in the class of univalent functions, providing sharp bounds and conjecturing the maximal radius for univalence within certain subclasses.

Contribution

It offers new sharp results on the univalence radius of $f(z)/f'(z)$ for subclasses of univalent functions and proposes a conjecture for the maximal univalence radius.

Findings

If $f$ is in ${ m extbf{S}}$, then $F$ is univalent in $|z|<r_6 \\approx 0.360794$.

The paper establishes several sharp and non-sharp bounds for the univalence radius.

Conjecture: the maximal univalence radius is $oxed{ extstyle rac{ extbf{ extup{1}}}{ extbf{ extup{( extbf{ extup{sqrt(2)}}-1)}}}}$.

Abstract

Let ${\mathcal S}$ denote the family of all univalent functions $f$ in the unit disk $\ID$ with the normalization $f(0)=0= f'(0)-1$. There is an intimate relationship between the operator $P_f(z)=f(z)/f'(z)$ and the Danikas-Ruscheweyh operator $T_f:=\int_{0}^{z}(tf'(t)/f(t))\,dt$. In this paper we mainly consider the univalence problem of $F=P_f$, where $f$ belongs to some subclasses of ${\mathcal S}$. Among several sharp results and non-sharp results, we also show that if $f\in {\mathcal S}$, then $F \in {\mathcal U}$ in the disk $|z|<r$ with $r\leq r_6\approx 0.360794$ and conjecture that the upper bound for such $r$ is $\sqrt{2}-1$.