Radius of Univalence of Certain Combination of Univalent and Analytic Functions

M. Obradovic ; S. Ponnusamy ; N. Tuneski

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 2

Résumé

Let $\mathcal{A}$ denote the family of all analytic functions $f$ in the unit disk $D$ with the normalization $f(0)=0= f'(0)-1$. Define $\mathcal{S} = \{ f \in \mathcal{A}: \, f ~\mbox{is univalent in } D \}$, $\mathcal{U} = \{ f \in \mathcal{A} :\, \big |f'(z)\left (z/f(z) \right )^{2}-1\big | 1 ~\mbox{ for $z\in D$} \}$, and $ \mathcal{P}(1/2)= \{f\in \mathcal{A}:\, {\rm Re\,}(f(z)/z)>1/2$ $ ~\mbox{ for $z\in D$} \}.$ In this paper, we determine the radius of univalency of $F(z)=zf(z)/g(z)$ whenever $f\in \mathcal{ S} $ or $\mathcal{U}$, and $g\in \mathcal{S} $ or $\mathcal{P}(1/2)$. Based on our investigations, we conjecture that $F$ is univalent in the disk $|z|1/3$ whenever $f\in \mathcal{S}$ and $g\in\mathcal{ P}(1/2)$. We also conjecture that $F$ is univalent in the disk $|z|\sqrt{5}-2$ whenever both $f$ and $g$ are in $\mathcal{S}$.

Détail

Classification : 30C45.