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

Abstract

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

11 pages

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