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

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 $\sqrt{2}-1$.