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  Volume 10, Issue 4, Article 113
An Extension of the Region of Variability of a Subclass of Univalent Functions

    Authors: Sukhwinder Singh, Sushma Gupta, Sukhjit Singh,  
    Keywords: Analytic Function, Univalent function, Starlike function, Differential subordination.  
    Date Received: 03/05/2009  
    Date Accepted: 05/11/2009  
    Subject Codes:

30C80, 30C45.

    Editors: Sever S. Dragomir,  

We show that for $\alpha \in (0,2],$ if $f\in \mathcal{A}$ with $f^{\prime }(z)\neq 0, z\ in \mathbb{E},$ satisfies the condition

\begin{displaymath} (1-\alpha)f^{\prime }(z)+\alpha( 1+ \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}) \prec F(z), \end{displaymath}

then $f$ is univalent in $\mathbb{E},$ where $F$ is the conformal mapping of the unit disk $\mathbb{E}$ with $F(0)=1$ and
\begin{displaymath} F(\mathbb{E})={\mathbb{C}}\setminus \{w\in {\mathbb{C}... ...a,  \vert\Im  w\vert \geq \sqrt{\alpha (2-\alpha)} \}. \end{displaymath}

Our result extends the region of variability of the differential operator
\begin{displaymath} (1-\alpha)f^{\prime }(z)+\alpha ( 1+ \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}), \end{displaymath}

implying univalence of $f\in \mathcal{A}$ in $\mathbb{E},$ for $0<\alpha \leq 2$.

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