Where is $f(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)$ univalent?

Let $\mathcal{S}$ denote the family of all univalent functions $f$ in the unit disk $\text{ID}$ with the normalization $f(0)=0=f^{\mathrm{\prime }}(0)-1$. There is an intimate relationship between the operator $P_f(z)=f(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)$ and the Danikas-Ruscheweyh operator $T_f:={\int }_0^z(t{f}^{\mathrm{\prime }}(t)\mathrm{/}f(t))dt$. 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 $\mathrm{\mid }z\mathrm{\mid }<r$ with $r\le r_6\approx 0.360794$ and conjecture that the upper bound for such $r$ is $\sqrt{2}-1$.