IJPAM: Volume 73, No. 4 (2011)

ON RADIUS PROBLEMS IN
THE CLASS OF UNIVALENT FUNCTIONS

Ahmed Amer$^1$, Maslina Darus$^2$
$^{1,2}$School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
Bangi, Selangor, 43600, MALAYSIA


Abstract. Let $SL^*(\beta)$ denote the class of all analytic functions $f$ in the unit disc $\mathbb{U}$ with the normalization $f(0)=f'(0)-1=0,$ and satisfying the condition

\begin{displaymath}\left\vert\left(\frac {zf'(z)}{f(z)}\right)^2-(1-\beta)\right\vert<1-\beta
\quad ,(z\in \mathbb{U})\}.\end{displaymath}

Thus, $\frac {zf'(z)}{f(z)}$ is the interior of the right half of the lemniscate of Bernoulli $\gamma:(x^2+y^2)^2-2(1-\beta)(x^2-y^2)=0.$ The radii of $\beta-$convexity, $\beta-$starlikeness (and some of others) for $f \in SL^*(\beta)$ are determined.

Received: August 21, 2011

AMS Subject Classification: 30C45

Key Words and Phrases: analytic functions, convex functions, starlike functions, $k$-starlike functions, strongly starlike functions

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 73
Issue: 4