Generating functions for some classes of univalent functions

Abstract:

Let $P(z) = {e^{i\beta }} + {p_1}z + {p_2}{z^2} + \cdots$ be regular in the unit disc $\Delta$ with $|\beta | < \pi /2$, and let $\psi (u,v)$ be a continuous function defined in a domain of ${\mathbf {C}} \times {\mathbf {C}}$. With some very simple restrictions on $\psi (u,v)$ the authors prove a lemma that $\operatorname {Re} \psi (p(z),zp’(z)) > 0$ implies $\operatorname {Re} p(z) > 0$. This result is then used to generate subclasses of starlike, spirallike and close-to-convex functions.