Generating functions for some classes of univalent functions

Abstract

Let P ( z ) = e i β + p 1 z + p 2 z 2 + ⋯ P(z) = {e^{i\beta }} + {p_1}z + {p_2}{z^2} + \cdots be regular in the unit disc Δ \Delta with | β | > π / 2 |\beta | > \pi /2 , and let ψ ( u , v ) \psi (u,v) be a continuous function defined in a domain of C × C {\mathbf {C}} \times {\mathbf {C}} . With some very simple restrictions on ψ ( u , v ) \psi (u,v) the authors prove a lemma that Re ⁡ ψ ( p ( z ) , z p ′ ( z ) ) > 0 \operatorname {Re} \psi (p(z),zp’(z)) > 0 implies Re ⁡ p ( z ) > 0 \operatorname {Re} p(z) > 0 . This result is then used to generate subclasses of starlike, spirallike and close-to-convex functions.