Radii properties for subclasses of convex functions (Q1910829)

Let \(\mathcal D\) denote the set of functions \(f\) that are analytic in the open unit disk and satisfy \(f(0)=0\), \(f'(0)=1\) and \(|zf''(z)|\leq\text{Re }f'(z)\) for \(|z|<1\). \(\mathcal D\) was introduced by \textit{St. Ruscheweyh} [SIAM J. Math. Anal. 19, 1442-1449 (1988; Zbl 0661.30012)] and each function in \(\mathcal D\) is a convex mapping. The authors determine the largest value of \(\beta\) such that \(f(\beta z)/\beta\in {\mathcal D}\) when \(f\) is convex of order \(\alpha\) \((\alpha\geq 0)\) as well as when \(f\) satisfies \(\text{Re }f'(z)>\alpha\) for \(|z|<1\) \((0\leq\alpha<1)\). The arguments depend on knowing the extreme points of the closed convex hulls of the two classes.