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Abstract

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 $U_0(\lambda)$.