On the radius of convexity for a class of conformal maps

V. Karunakaran ¹   ; K. Bhuvaneswari ²

¹ School of Mathematics Madurai Kamaraj University Madurai 625021, India
² chool of Mathematics Madurai Kamaraj University Madurai 625021, India

Colloquium Mathematicum, Tome 109 (2007) no. 2, pp. 251-256

Résumé

Let $\mathcal{A}$ denote the class of all analytic functions $f$ in the open unit disc $\mathbb{D}$ in the complex plane satisfying $f(0)=0,f'(0)=1$. Let $U(\lambda)\ (0 \lambda \leq 1)$ denote the class of functions $f \in \mathcal{A}$ for which $$ \left|\left(\frac{z}{f(z)}\right)^2f'(z) -1 \right| \lambda \quad \mbox{for} \ z \in \mathbb{D}. $$ The behaviour of functions in this class has been extensively studied in the literature. In this paper, we shall prove that no member of $U_{0}(\lambda) = \left\{f\in U(\lambda): f' '(0) =0 \right\}$ is convex in $\mathbb{D}$ for any $ \lambda $ and obtain a lower bound for the radius of convexity for the family $U_0(\lambda)$. These results settle a conjecture proposed in the literature negatively. We also improve the existing lower bound for the radius of convexity of the family $U_0(\lambda)$.

Détail

DOI : 10.4064/cm109-2-7

Keywords: mathcal denote class analytic functions unit disc mathbb complex plane satisfying lambda lambda leq denote class functions mathcal which frac right right lambda quad mbox mathbb behaviour functions class has extensively studied literature paper shall prove member lambda lambda right convex mathbb lambda obtain lower bound radius convexity family lambda these results settle conjecture proposed literature negatively improve existing lower bound radius convexity family lambda