Geometric properties of the Cesàro means (Q1205491)

Set \(S_ \gamma^*\) denote the class of functions \(f(z)=z+\dots\) analytic in the unit disk \(\mathbb{D}\) such that \(\text{Re}[zf'(z)/f(z)]>\gamma\). The author shows that, if \(\alpha\geq 1\), \[ {{n+\alpha-1} \choose {n-1}}^{-1} \sum_{k=1}^ n {{n+\alpha-k} \choose {n-k}} {{2\gamma-2} \choose {k-1}}z^ k \in S_ \gamma^* \] for \(\gamma=(3-\alpha)/2\) but not for any smaller \(\gamma\). This result implies that, for any function \(f(z)=z+\dots\) analytic in \(\mathbb{D}\), \[ f\in S_{(3-\alpha)/2}^* \iff \forall n:\quad S_ n^ \alpha (z,f)\in S_{(3-\alpha)/2}^* \] where \(S_ n^ \alpha(z,f)\) is the Cesàro mean. It is conjectured that \[ \sigma_ n^ \alpha (z)={{n+\alpha} \choose n}^{-1} \sum_{k=0}^ n {{n+\alpha-k} \choose {n-k}}z^ k \] satisfies the subordination relation \(\sigma_ n^ \alpha(z)\prec \sigma_ n^ \beta (z)\) for \(2\leq\beta\leq\alpha\).