LOWER BOUND TO A PROBLEM OF MOCANU ON DIFFERENTIAL SUBORDINATION

Let $s^*$ denote the family of starlike mappings in the unit disc $\Delta$. Let $\mathcal{R}(\alpha, \beta)$ denote the family of normalized analytic functions in $\Delta$ satisfying the condition Re$(f'(z)+\alpha f''(z))>\beta$, $z \in\Delta$ for some $\alpha > 0$. In this note, among other things, we give a lower bound to the problem of Mocanu aimed at determining $\inf\{\alpha : \mathcal{R}(\alpha,0) \subset S^*\}$.